SOME HOMOLOGICAL PROPERTIES OF CATEGORY $\boldsymbol {\mathcal {O}}$ FOR LIE SUPERALGEBRAS

Abstract

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\Delta (\lambda )$ to be such that every nonzero homomorphism from another Verma supermodule to $\Delta (\lambda )$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak {pe} (n)$ and, furthermore, to reduce the problem of description of $\mathrm {Ext}^1_{\mathcal O}(L(\mu ),\Delta (\lambda ))$ for $\mathfrak {pe} (n)$ to the similar problem for the Lie algebra $\mathfrak {gl}(n)$ . Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\mathcal O^{\mathfrak {p}}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak {pe} (n)$ and the orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$ .

1 Introduction

Contemporary representation theory uses a wide range of homological methods. This makes understanding homological properties of algebraic structures an interesting and important problem.

In Lie theory, one of the most important representation theoretic objects of study is the Bernstein–Gelfand–Gelfand (BGG) category $\mathcal {O}$ associated to a fixed triangular decomposition of a semisimple finite dimensional complex Lie algebra, introduced in [BGG1, BGG2]. This category has a number of remarkable properties and applications; see [Hu] and references therein. The study of various homological invariants of category $\mathcal {O}$ goes back to the original paper [BGG2] which determines its global dimension. Some time ago, the second author started a series of papers [Ma1, Ma2] aimed at a systematic study of homological invariants of various classes of structural modules in category $\mathcal {O}$ . This series was subsequently continued with [CoM1, CoM2, KMM1] covering a wider class of questions and objects and also extending the results to Rocha-Caridi’s parabolic generalization of $\mathcal {O}$ from [RC].

Both category $\mathcal {O}$ and its parabolic generalization, can be defined, in a natural way, in the setup of Lie superalgebras; see [AM, CW1, Ka, Ma3, Mu]. The objects obtained in this way are very interesting and highly nontrivial; see [CW1, Mu] and references therein. Therefore, it is natural to study homological invariants for category $\mathcal {O}$ for Lie superalgebras. The first step here would be to find out which of the classical results (that is, results for Lie algebras) can be generalized to the supersetting and how. This is one of the motivations behind some parts of the present paper.

A recent paper [KMM2] discovered an interesting connection between bigrassmannian permutations and cokernels of inclusions of Verma modules for the Lie algebra $\mathfrak {sl}_n$ . The second motivation of this paper is to investigate to which extent this result generalizes to the supersetting. This is a nontrivial challenge as it is very well known that, unlike the classical case, homomorphisms between Verma supermodules over Lie superalgebras are often not injective.

To our surprise, it turns out that both for the homological properties of structural supermodules and for the description of cokernels of inclusions of Verma supermodules, it is possible to obtain fairly complete results for the periplectic Lie superalgebra $\mathfrak {pe} (n)$ . This is the main aim of the present paper.

Let us now briefly describe the structure of the paper and the main results. In Section 2, we collect preliminaries on classical Lie superalgebras which are relevant for the rest of the paper.

In Section 3, we describe Verma supermodules for classical Lie superalgebras of type I having the property that any homomorphism from any other Verma supermodule to them is injective. As an application, we prove that every nonzero homomorphism between Verma supermodules for $\mathfrak {pe} (n)$ is an embedding.

In Section 4, we describe the socle of the cokernel of a nonzero homomorphism between Verma supermodules for $\mathfrak {pe} (n)$ . We manage to reduce the problem for $\mathfrak {pe} (n)$ to the corresponding problem for $\mathfrak {gl}(n)$ , where the main result of [KMM2] applies.

Section 5 is devoted to the study of possible connections between homological invariants for a Lie superalgebra and the corresponding homological invariants for its even part, which is a Lie algebra. In particular, we relate the finitistic dimension of the category of supermodules for a classical Lie superalgebra to the global dimension of the category of modules for its even part. In Sections 5.3 and 5.4, we focus on projective and injective dimensions of structural modules in an arbitrary parabolic category $\mathcal O^{\mathfrak {p}}$ . In Section 5.5, we introduce the notion of the associated variety and discuss its relation to projective dimension.

Finally, in Section 6, we investigate the projective dimensions of structural modules in the parabolic category $\mathcal O^{\mathfrak {p}}$ for $\mathfrak {pe} (n)$ and the orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$ .

2 Preliminaries

Throughout, we let $\mathfrak g = \mathfrak g_{\overline 0}\oplus \mathfrak g_{{\bar 1}}$ be a finite-dimensional classical Lie superalgebra. This means that $\mathfrak g_{\bar 0}$ is a reductive Lie algebra and $\mathfrak g_{\bar 1}$ is a completely reducible $\mathfrak g_{\bar 0}$ -module under the adjoint action.

For a given classical Lie superalgebra, we fix triangular decompositions

(2-1) $$ \begin{align} & \mathfrak g_{\bar 0} =\mathfrak n_{\bar 0} \oplus \mathfrak h_{\bar 0} \oplus \mathfrak n^-_{\bar 0} \quad\text{and}\quad \mathfrak g =\mathfrak n \oplus \mathfrak h \oplus \mathfrak n^- \end{align} $$

of $\mathfrak g_{\bar 0}$ and of $\mathfrak {g}$ , respectively, in the sense of [Ma3, Section 2.4]. Here, $\mathfrak b_{\bar 0}:=\mathfrak n_{\bar 0} \oplus \mathfrak h_{\bar 0}$ and $\mathfrak b:=\mathfrak n \oplus \mathfrak h$ are Borel subalgebras in $\mathfrak g_{\bar 0}$ and $\mathfrak g$ , respectively. In the present paper, we assume that the Cartan subalgebra $\mathfrak h$ is purely even, that is $\mathfrak h=\mathfrak h_{\bar 0}$ . We denote by $ \mathfrak g=\bigoplus _{\alpha \in \mathfrak {h}^\ast }\mathfrak g^\alpha $ the root space decomposition of $\mathfrak g$ with respect to $\mathfrak h^\ast $ . We denote by W the Weyl group of $\mathfrak g_{\bar 0}$ . The usual dot-action of W on $\mathfrak h^\ast $ is defined as

$$ \begin{align*}w\cdot \lambda := w(\lambda+\rho_{\bar 0})-\rho_{\bar 0},\quad\text{for any }w\in W,~\lambda \in \mathfrak{h}^\ast,\end{align*} $$

where $\rho _{\bar 0}$ is half of the sum of all positive roots of $\mathfrak g_{\bar 0}$ . For a given weight $\mu \in \mathfrak {h}^\ast $ , we let $W_\mu $ denote the stabilizer of $\mu $ under the dot-action of W. Let $\ell : W\rightarrow \mathbb {Z}_{\geq 0}$ denote the usual length function.

Finally, a weight is called integral, dominant or anti-dominant if it is integral, dominant or antidominant as a $\mathfrak g_{\bar 0}$ -weight, respectively.

Finally, we denote by $\Phi ^+, \Phi ^+_{\bar 0}$ and $\Phi ^+_{\bar 1}$ the sets of positive roots, even positive roots and odd positive roots with respect to the triangular decomposition (2-1), respectively. In other words, these are exactly the nonzero weights of $\mathfrak b$ , $\mathfrak b_{\bar 0}$ and $\mathfrak b_{\bar 1}$ , respectively.

2.1 BGG category $\mathcal O$

Associated to the triangular decomposition (2-1) above, the BGG category $\mathcal O $ (respectively $\mathcal O_{\bar 0}$ ) is defined as the full subcategory of the category of finitely generated $\mathfrak {g}$ -modules (respectively $\mathfrak g_{\bar 0}$ -modules) which are semisimple over $\mathfrak h$ and locally $U(\mathfrak {n}^+)$ -finite (respectively $U(\mathfrak {n}^+_{\bar 0})$ -finite); see [BGG2, Hu].

For $\lambda \in \mathfrak {h}^\ast $ , the corresponding Verma modules over $\mathfrak g_{\bar 0}$ and $\mathfrak g$ are defined as

$$ \begin{align*}\Delta_{{\bar 0}}(\lambda)=U(\mathfrak{g}_{{\bar 0}})\otimes_{U(\mathfrak{b}_{{\bar 0}})}\mathbb{C}_{\lambda}\quad\text{and}\quad\Delta(\lambda) := U(\mathfrak g)\otimes_{U(\mathfrak b)} \mathbb{C}_{\lambda},\end{align*} $$

respectively. Here, $\mathbb {C}_{\lambda }$ is the one-dimensional module over the Borel subalgebra corresponding to $\lambda $ . The unique simple quotients of $\Delta _{\bar 0}(\lambda )$ and $\Delta (\lambda )$ are denoted by $L_{\bar 0}(\lambda )$ and $L(\lambda )$ , respectively. The corresponding dual Verma modules $\nabla _{\bar 0}(\lambda )$ and $\nabla (\lambda )$ are similarly defined as coinduced modules; see [Hu, Section 3.2] and [CCC, Definition 3.2] for more details. We note that $L(\lambda ) \hookrightarrow \nabla (\lambda ).$ Also, we denote by $T_{\bar 0}(\lambda )$ and $T(\lambda )$ the tilting modules of highest weight $\lambda $ in $\mathcal O_{\bar 0}$ and $\mathcal O$ , respectively; see [Ma3, Section 4.3] and [CCC, Theorem 3.5] for more details.

Let us denote by $\operatorname {Res}\nolimits :=\operatorname {Res}\nolimits _{\mathfrak g_{\bar 0}}^{\mathfrak {g}}$ , $\operatorname {Ind}\nolimits :=\operatorname {Ind}\nolimits _{\mathfrak g_{\bar 0}}^{\mathfrak {g}}$ and $\mathrm {Coind}:=\mathrm {Coind}_{\mathfrak g_{\bar 0}}^{\mathfrak {g}}$ the corresponding restriction, induction and coinduction functors, respectively. It is well known; see [BF, Theorem 2.2] and [Gor1], that there is an isomorphism of functors

$$ \begin{align*}\operatorname{Ind}\nolimits \cong \mathrm{Coind}( \Lambda^{\text{top}}\mathfrak g_{\bar 1}\otimes-).\end{align*} $$

2.2 Lie superalgebras of type I

A classical Lie superalgebra $\mathfrak g = \mathfrak g_{\overline 0}\oplus \mathfrak g_{{\bar 1}}$ is said to be of type I if $\mathfrak g$ has a compatible $\mathbb {Z}$ -grading $\mathfrak g= \mathfrak g_{-1}\oplus \mathfrak g_0 \oplus \mathfrak g_{1}$ such that $\mathfrak g_0 =\mathfrak g_{\overline 0}$ and $\mathfrak g_{\bar 1} = \mathfrak g_1 \oplus \mathfrak g_{-1}$ , where $\mathfrak g_{\pm 1}$ are $\mathfrak g_{\overline 0}$ -submodules of $\mathfrak g_{\bar 1}$ with $[\mathfrak g_{1},\mathfrak g_{ 1}] =[\mathfrak g_{-1},\mathfrak g_{-1}] =0$ . We set $\mathfrak g_{\geq 0}: = \mathfrak g_0 \oplus \mathfrak g_1$ and $\mathfrak g_{\leq 0}: = \mathfrak g_0 \oplus \mathfrak g_{-1}$ .

2.2.1 Triangular decompositions for $\mathfrak g$ of type I

For classical Lie superalgebras $\mathfrak g$ of type I considered in the present paper, we always assume that the triangular decomposition in (2-1) has the property

(2-2) $$ \begin{align} &\mathfrak b = \mathfrak b_{\bar 0} \oplus \mathfrak g_1, \end{align} $$

that is, $\mathfrak b_{\bar 1}=\mathfrak {g}_1$ .

In particular, we are interested in the following superalgebras from Kac’s list (see [Ka]):

(2-3) $$ \begin{align} &(\text{Type}\ {\bf A}): ~\mathfrak{gl}(m|n), \mathfrak{sl}(m|n) \text{ and } \mathfrak{sl}(n|n)/\mathbb{C}I_{n|n}; \end{align} $$

(2-4) $$ \begin{align} (\text{Type}\ {\bf C}) &: ~\mathfrak{osp}(2|2n); \nonumber\\ (\text{Type}\ {\bf P}) &: ~\mathfrak{pe} (n)\text{ and } [\mathfrak{pe} (n), \mathfrak{pe} (n)]; \kern37pt \end{align} $$

where $m>n \geq 1$ are integers (see, for example, [Mu, Ch. 2 and 3] and [CW1, Section 1.1]). See Sections 3.2 and 6.1 for more details on $\mathfrak {pe} (n)$ and $\mathfrak {osp}(2|2n)$ .

2.2.2 Induced and coinduced modules

For a given object $V \in \mathcal O_{\bar 0}$ , we extend V trivially to a $\mathfrak g_{\geq 0}$ -module and define the Kac module

$$ \begin{align*}K(V):=\operatorname{Ind}\nolimits_{\mathfrak g_{\geq 0}}^{\mathfrak g}V.\end{align*} $$

This defines an exact functor $K({}_-): \mathcal O_{\bar 0} \rightarrow \mathcal O$ which is called the Kac functor. For $\lambda \in \mathfrak {h}^\ast $ , we define $K(\lambda ) := K(L_{\bar 0}(\lambda ))$ . Also, we note that $\Delta (\lambda ) \cong K(\Delta _{\bar 0}(\lambda ))$ and $\nabla (\lambda ):= \text {Coind}_{\mathfrak g_{\leq 0}}^{\mathfrak g} (\nabla _{\bar 0}(\lambda )).$

3 Homomorphisms between Verma supermodules

In this section, we assume that:

1. (1) $\mathfrak g$ is a classical Lie superalgebra of type I such that the triangular decomposition in (2-1) satisfies the condition in (2-2);

2. (2) there is an element $d\in \mathcal Z(\mathfrak g_{\bar 0})$ and a nonzero scalar $D\in {\mathbb C}$ such that

   $$ \begin{align*}[d,x]=Dx,\quad \text{for any} ~ x\in \mathfrak g_{-1}.\end{align*} $$

One can check that the superalgebras $\mathfrak {gl}(m|n)$ , $\mathfrak {osp}(2|2n)$ and $\mathfrak {pe}(n)$ from the list (2-3)–(2-4) satisfy conditions (1) and (2). We refer to [CM, Section 4.2] for a description of the elements d.

One of the goals in this subsection is to give a necessary and sufficient condition for the homomorphism

$$ \begin{align*}K({}_-): \text{Hom}_{\mathcal O_{\bar 0}}(\Delta_{\bar 0}(\mu),\Delta_{\bar 0}(\lambda)) \longrightarrow \text{Hom}_{\mathcal O}(\Delta(\mu),\Delta(\lambda)),\end{align*} $$

where $K({}_-)$ is the Kac functor defined in Section 2.2 to be an isomorphism.

3.1 Kac functor and homomorphisms between Verma supermodules

Using the same argument as in the proof of [CP, Lemma 4.3], we have the following useful lemma.

Lemma 1. Let $\mu , \nu \in \mathfrak {h}^\ast $ be integral weights such that $[K(\mu ):L(\nu )]>0$ and $K(\nu )=L(\nu )$ . Then $\mu =\nu $ .

The following is our first main result.

Theorem 2. Let $\lambda \in \mathfrak {h}^\ast $ be integral. Then the following conditions are equivalent.

1. (1) The Kac module $K(\underline \lambda )$ is simple, where $\underline \lambda \in W\cdot \lambda $ is antidominant.

2. (2) For any $\mu \in \mathfrak {h}^\ast $ , every nonzero homomorphism $f:\Delta (\mu )\rightarrow \Delta (\lambda )$ is an embedding.

3. (3) For any $\mu \in \mathfrak {h}^\ast $ , the Kac functor $K({}_-)$ gives rise to an isomorphism,

   $$ \begin{align*} &K({}_-): \mathrm{Hom}_{\mathcal O_{\bar 0}}(\Delta_{\bar 0}(\mu),\Delta_{\bar 0}(\lambda)) \xrightarrow{\cong} \mathrm{Hom}_{\mathcal O}(\Delta(\mu),\Delta(\lambda)).\end{align*} $$

Proof. In the proof, we regard $\Delta _{\bar 0}(\lambda )$ and $\Delta _{\bar 0}(\mu )$ as direct summands of $\operatorname {Res}\nolimits \Delta (\lambda )$ and $\operatorname {Res}\nolimits \Delta (\mu )$ , respectively.

We first prove $(1)\Rightarrow (3)$ . Using the exactness of Kac functor, the characters of $\Delta (\lambda )$ and $\Delta (\mu )$ are given as follows:

$$ \begin{align*} &\text{ch}\Delta(\lambda) = \sum_{\zeta} [\Delta_{\bar 0}(\lambda): L_{\bar 0}(\zeta)] \text{ch}K(\zeta),\quad \text{ch}\Delta(\mu) = \sum_{\zeta} [\Delta_{\bar 0}(\mu): L_{\bar 0}(\zeta)] \text{ch}K(\zeta). \end{align*} $$

This implies that there exists $\zeta \in W\cdot \mu $ such that $[K(\zeta ):L(\underline \lambda )]>0$ because f is nonzero and soc $(\Delta (\lambda )) =L(\underline \lambda )$ by [CM, Theorem 51].

Because $d\in \mathcal Z(\mathfrak g_{\bar 0})$ , the element d acts on $\Delta _{\bar 0}(\lambda )$ and $\Delta _{\bar 0}(\mu )$ via some scalars, say $c_\lambda $ and $c_\mu $ , respectively. From Lemma 1, we conclude that $\zeta =\underline \lambda $ because $[K(\zeta ):L(\underline \lambda )]>0$ and $K(\underline \lambda )=L(\underline \lambda )$ . Therefore, $\mu \in W\cdot \lambda $ and hence $c_\lambda =c_\mu $ .

Applying the restriction functor to f, we obtain a $\mathfrak {g}_{\bar 0}$ -homomorphism

$$ \begin{align*}\operatorname{Res}\nolimits(f): \Lambda^{\bullet} \mathfrak g_{-1} \otimes \Delta_{\bar 0}(\mu) \rightarrow \Lambda^{\bullet} \mathfrak g_{-1}\otimes \Delta_{\bar 0}(\lambda).\end{align*} $$

We now consider the eigenspaces with respect to the operator d. Observe that for any $k\geq 0$ , the subspace $\Lambda ^{k} \mathfrak g_{-1}\otimes \Delta _{\bar 0}(\mu )$ of $\operatorname {Res}\nolimits \Delta (\mu )$ is the eigenspace for d corresponding to the eigenvalue $c_\mu +kD$ . Similarly, the subspace $\Lambda ^{k} \mathfrak g_{-1}\otimes \Delta _{\bar 0}(\lambda )$ of $\operatorname {Res}\nolimits \Delta (\lambda )$ is the eigenspace for d corresponding to the eigenvalue $c_\lambda +kD$ . Therefore, we have

(3-1) $$ \begin{align} 0\neq \operatorname{Res}\nolimits(f)(\Delta_{\bar 0}(\mu)) \subseteq \Delta_{\bar 0}(\lambda) \end{align} $$

because $f\neq 0$ and $c_\lambda =c_\mu .$ Consequently, we may conclude that

(3-2) $$ \begin{align} \operatorname{Res}\nolimits(f){|}_{\Delta_{\bar 0}(\mu)}: \Delta_{\bar 0}(\mu) \rightarrow \Delta_{\bar 0}(\lambda) \end{align} $$

is an embedding by [Di, Theorem 7.6.6].

Let K be the kernel of f. If we assume that f is not an embedding, then we have $K \supseteq $ soc $(\Delta (\mu ))$ by [CCM, Theorem 51]. This gives

$$ \begin{align*} &K \supseteq \text{soc}(\Delta(\mu)) = U(\mathfrak g)\cdot (\Lambda^{\text{top}}\mathfrak g_{-1}\otimes \text{soc}(\Delta_{\bar 0}(\mu))), \end{align*} $$

which means that

$$ \begin{align*}\Lambda^{\text{top}}\mathfrak g_{-1}\otimes \operatorname{Res}\nolimits(f)(\text{soc} (\Delta_{\bar 0}(\mu)))=f(\Lambda^{\text{top}}\mathfrak g_{-1}\otimes \text{soc}(\Delta_{\bar 0}(\mu))) =0,\end{align*} $$

a contradiction to (3-1). Note that this proves the direction $(1)\Rightarrow (2)$ . We proceed, however, with the proof of the direction $(1)\Rightarrow (3)$ .

Observe that the Kac functor gives rise to a monomorphism

$$ \begin{align*}K({}_-): \text{Hom}_{\mathcal O_{\bar 0}}(\Delta_{\bar 0}(\mu),\Delta_{\bar 0}(\lambda)) \rightarrow \text{Hom}_{\mathcal O}(\Delta(\mu),\Delta(\lambda)).\end{align*} $$

It suffices to show that for given nonzero $f,g\in \text {Hom}_{\mathcal O}(\Delta (\mu ),\Delta (\lambda ))$ , there exists $\alpha \in {\mathbb C}$ such that $f=\alpha g.$ By (3-2), both f and g restrict to nonzero $\mathfrak g_{\bar 0}$ -homomorphisms $\operatorname {Res}\nolimits (f){|}_{\Delta _{\bar 0}(\mu )}$ and $\operatorname {Res}\nolimits (g){|}_{\Delta _{\bar 0}(\mu )}$ from $\Delta _{\bar 0}(\mu )$ to $\Delta _{\bar 0}(\lambda )$ , respectively. From [Di, Theorem 7.6.6], we have that there exists $\alpha \in {\mathbb C}$ such that

$$ \begin{align*}\operatorname{Res}\nolimits(f){|}_{\Delta_{\bar 0}(\mu)} =\alpha \operatorname{Res}\nolimits(g){|}_{\Delta_{\bar 0}(\mu)}.\end{align*} $$

This proves the part $(1)\Rightarrow (3)$ because $\Delta (\mu )$ is generated by $\Delta _{\bar 0}(\mu )$ as a $\mathfrak g$ -module.

Note that the implication $(3)\Rightarrow (2)$ is clear. It remains to prove the implication $(2)\Rightarrow (1)$ , so we now assume $(2)$ . We set $L(\nu ):=\text {soc}(K(\underline \lambda ))$ . Then we have the map $\Delta (\nu ) \twoheadrightarrow L(\nu ) \hookrightarrow K(\underline \lambda )= \Delta (\underline \lambda )$ . Because this map is an embedding by $(2)$ , it follows that $\Delta (\nu ) = L(\nu ) \hookrightarrow K(\underline \lambda )= \Delta (\underline \lambda )$ . By Lemma 1, from $[K(\underline \lambda ):L(\nu )]\neq 0$ and $K(\nu ) =L(\nu )$ , we deduce that $\underline \lambda = \nu $ . This gives $K(\underline \lambda ) =L(\underline \lambda )$ and completes the proof.

3.2 Example: the periplectic Lie superalgebras $\mathfrak {pe} (n)$

For positive integers $m,n$ , the general linear Lie superalgebra $\mathfrak {gl}(m|n)$ can be realized as the space of $(m+n) \times (m+n)$ complex matrices

$$ \begin{align*} \left( \begin{array}{cc} A & B\\ C & D\\ \end{array} \right), \end{align*} $$

where $A,B,C$ , and D are $m\times m, m\times n, n\times m$ , and $n\times n$ matrices, respectively. The Lie bracket of $\mathfrak {gl}(m|n)$ is given by the super commutator. Let $E_{ab}$ , for $1\leq a,b \leq m+n$ , be the elementary matrix in $\mathfrak {gl}(m|n)$ . Its $(a,b)$ -entry is equal to $1$ and all other entries are $0$ .

The standard matrix realization of the periplectic Lie superalgebra $\mathfrak {pe} (n)$ inside the general linear Lie superalgebra $\mathfrak {gl}(n|n)$ is given by

(3-3) $$ \begin{align} \mathfrak g= \mathfrak{pe}(n):= \left\{ \left( \begin{array}{cc} A & B\\ C & -A^t\\ \end{array} \right)\bigg\| ~ A,B,C\in {\mathbb C}^{n\times n},~B^t=B,\ \text{ and }\ C^t=-C \right\}. \end{align} $$

Throughout the present paper, we fix the Cartan subalgebra $\mathfrak h= \mathfrak h_{\bar 0} \subset \mathfrak g_{\overline 0}$ consisting of diagonal matrices. We denote the dual basis of $\mathfrak h^*$ by $\{\epsilon _1, \epsilon _2, \ldots , \epsilon _n\}$ with respect to the standard basis of $\mathfrak h$ defined as

$$ \begin{align*} \{H_i:=E_{i,i}-E_{n+i,n+i}\mid 1\leq i \leq n \}\subset \mathfrak{pe} (n), \end{align*} $$

where $E_{a,b}$ denotes the $(a,b)$ -matrix unit for $1\leq a,b \leq 2n$ . The sets $\Phi _{\bar 0}$ , $\Phi _{\bar 1}^\pm $ , and $\Phi _{\bar 1}$ of even, odd positive, odd negative, and odd roots are, respectively, given by

$$ \begin{align*} &\Phi_{\bar 0}=\{\epsilon_i-\epsilon_j\mid 1\le i\not=j\le n\}, \nonumber\\ &\Phi^+_{\bar 1}=\{\epsilon_i+\epsilon_j\mid 1\le i\le j\le n\} ,\nonumber\\ &\Phi^-_{\bar 1}= \{- \epsilon_i-\epsilon_j\mid 1\le i< j\le n\},\nonumber\\ &\Phi_{\bar 1}:=\Phi_{\bar 1}^+\cup \Phi_{\bar 1}^-. \end{align*} $$

The space $\mathfrak h^\ast = \bigoplus _{i=1}^n{\mathbb C} \epsilon _i$ is equipped with a natural bilinear form $\langle \epsilon _i , \epsilon _j \rangle =\delta _{i,j}$ , for any $1\leq i,j \leq n$ . The Weyl group $W=\mathfrak S_n$ is the symmetric group acting on $\mathfrak {h}^\ast $ . For any $\alpha \in \Phi _{\bar 0}$ , we let $s_\alpha $ denote the corresponding reflection in W.

We fix the Borel subalgebra $\mathfrak {b}_{{\bar 0}}$ of $\mathfrak {g}_{{\bar 0}}=\mathfrak {gl}(n)$ consisting of matrices in (3-3) with $B=C=0$ and A upper triangular.

The subalgebras $\mathfrak g_1$ and $\mathfrak g_{-1}$ are given by

$$ \begin{align*} \mathfrak g_1:= \bigg\{\begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix}\bigg|B^t=B\bigg\}\quad\mbox{and}\quad \mathfrak g_{-1}:= \bigg\{\begin{pmatrix} 0 & 0 \\ C & 0 \end{pmatrix}\bigg|C^t=-C\bigg\}. \end{align*} $$

We note that $\Phi _{\bar 1}^+$ and $\Phi _{\bar 1}^-$ are the sets of weights of the $\mathfrak g_0$ -modules $\mathfrak g_1$ and $\mathfrak g_{-1}$ , respectively.

The Borel subalgebra is defined as $\mathfrak b:=\mathfrak b_{\bar 0} \oplus \mathfrak {g}_{1}$ , while the reverse Borel subalgebra $\mathfrak b^r$ is defined as $\mathfrak b^r:=\mathfrak b_{\bar 0} \oplus \mathfrak g_{-1}.$ It can be shown that $\mathfrak b^r$ is still a Borel subalgebra in the sense of [Ma3, Section 2.4] (see, for example, [CCC, Section 5]).

Furthermore, we let $L^{r}(\lambda )$ be the irreducible module in the corresponding ‘reversed’ category $\mathcal O$ of $\mathfrak b^r$ -highest weight $\lambda $ . We let $\lambda ^+\in \mathfrak {h}^\ast $ be uniquely determined by

(3-4) $$ \begin{align} &L(\lambda^+) =L^r(\lambda). \end{align} $$

It is known that the weight $\lambda ^+$ can be computed using odd reflections, and we refer to [PS1, Section 2.2] and [Mu, Theorem 3.6.10] for a treatment of odd reflections for the periplectic Lie superalgebras $\mathfrak {pe} (n)$ . In particular, the effect on the highest weight of a simple $\mathfrak {pe} (n)$ -module under odd reflection and inclusion was computed in [PS1, Lemma 1].

Finally, we set $ X:=\bigoplus _{i=1}^n\mathbb {Z}\epsilon _i$ ; $\omega _k:=\epsilon _1+\cdots +\epsilon _k$ for any $1\leq k\leq n$ ; and also $\eta :=(1-n)(\epsilon _{1}+\cdots +\epsilon _{n})$ . Then we have $\Lambda ^{\text {top}}\mathfrak g_{-1}\cong {\mathbb C}_{\eta }$ as a $\mathfrak g_{\bar 0}$ -module.

Let $\lambda , \mu \in \mathfrak {h}^\ast $ be two integer weights. If there are positive roots

$$ \begin{align*}\alpha_1,\alpha_2,\ldots, \alpha_k\in \Phi_{\bar 0}^+,\end{align*} $$

such that $\lambda =s_{\alpha _k} s_{\alpha _{k-1}}\cdots s_{\alpha _1}\cdot \mu $ and, for each $1\leq q \leq k-1$ , we have

$$ \begin{align*}\langle s_{\alpha_q}s_{\alpha_{q-1}}\cdots s_{\alpha_1}\cdot \mu,~ \alpha_{q+1} \rangle \geq 0, \end{align*} $$

then we write $\mu \uparrow \lambda $ ; see [Di, Section 7.6].

Proposition 3. Let $\mathfrak g=\mathfrak {pe} (n)$ and $\lambda \in \mathfrak {h}^\ast $ be an integral weight.

1. (1) For any integral weight $\mu \in \mathfrak {h}^\ast $ , we have that

   $$ \begin{align*}\mathrm{Hom}_{\mathcal O}(\Delta(\mu),\Delta(\lambda)) \cong \begin{cases} {\mathbb C} \quad&\text{if } \mu \uparrow \lambda,\\ 0\quad& \text{otherwise. } \end{cases}\end{align*} $$

2. (2) A weight vector $v\in \Delta (\lambda )$ satisfies $\mathfrak nv=0$ if and only if

   $$ \begin{align*}v=1\otimes v_0\in U(\mathfrak g) \otimes_{\mathfrak g_{\geq 0}} \Delta_{\bar 0}(\lambda) \equiv \operatorname{Res}\nolimits \Delta(\lambda),\end{align*} $$
   for some vector $v_0$ satisfying $\mathfrak n_{\bar 0} v_0=0$ .

Proof. We recall that $\mathfrak {pe} (n)$ admits a grading operator $d:=$ diag $(1,1,\ldots ,1)\in \mathfrak {g}_{\bar 0}$ which satisfies condition (1) at the beginning of Section 3. Also, it follows from [ChC, Lemma 5.11] that every weight satisfies assumption $(1)$ in Theorem 2. Therefore, the proof follows from Theorem 2 and [Hu, Theorems 4.2 and 5.1].

Remark 4. In [LLW], Liu et al. established a closed formula for a singular vector of weight $\lambda -\beta $ in the Verma module of highest weight $\lambda $ for Lie superalgebra $\mathfrak {gl}(m|n)$ when $\lambda $ is atypical with respect to an odd positive root $\beta $ . In particular, the authors proved the following identity:

(3-5) $$ \begin{align} &\text{Hom}(\Delta(\lambda -\beta),\Delta(\lambda))={\mathbb C}, \end{align} $$

where $\Delta $ terms denote Verma supermodules for $\mathfrak {g} =\mathfrak {gl}(m|n)$ . This identity was also known in [CW2] for the exceptional Lie superalgebra $D(2|1;\zeta )$ . By Proposition 3, it follows that the identity (3-5) fails to hold for $\mathfrak {pe} (n)$ .

Remark 5. The Borel subalgebras have been classified in [CCC, Section 5]. It would be interesting to determine for which other Borel subalgebras one has an analogue of Proposition 3. We note that Proposition 3 fails in the case of the reverse Borel subalgebra $\mathfrak b^r$ . To see this, we define Verma supermodules with respect to $\mathfrak b^r$ as

$$ \begin{align*}\Delta^r(\lambda):=U(\mathfrak g)\otimes_{\mathfrak b^r}{\mathbb C}_\lambda \cong \text{Ind}^{\mathfrak{g}}_{\mathfrak g_{\leq 0}}\Delta_{\bar 0}(\lambda),\end{align*} $$

which admit a filtration with subquotients $ \text {Ind}^{\mathfrak {g}}_{\mathfrak g_{\leq 0}}L_{\bar 0}(\zeta )$ , for $\zeta \in \mathfrak {h}^\ast $ . By [CM, Proposition 4.15], the modules $\text {Ind}^{\mathfrak {g}}_{\mathfrak g_{\leq 0}}L_{\bar 0}(\zeta )$ are never simple, for any $\zeta \in \mathfrak {h}^\ast $ . By a similar argument as used in the directions $(3)\Rightarrow (2)$ and $(2)\Rightarrow (1)$ in the proof of Proposition 9, it follows that Proposition 9 fails for these modules $\Delta ^r(\lambda )$ .

Remark 6. Consider a basic Lie superalgebra $\mathfrak g$ of type I, for instance, $\mathfrak g=\mathfrak {gl}(m|n)$ or $\mathfrak {osp}(2|2n)$ . It has been shown by Kac [Ka2] that a weight $\lambda $ satisfies assumption (1) in Theorem 2 if and only if $\lambda $ is typical. Because the notions of strongly typical and typical are identical in these cases (see, for example, [Gor2, Section 2.4] or [Gor3]), the statement of Theorem 2 follows from Gorelik’s theorem; see [Gor2, Theorems 1.3.1 and 1.4.1].

4 Socle of the cokernel of inclusions of Verma supermodules for $\mathfrak {pe} (n)$

In this section, we assume that $\mathfrak g=\mathfrak {pe} (n)$ . The goal of this section is to describe the socle of the cokernel of an arbitrary nonzero homomorphism between Verma $\mathfrak {pe} (n)$ -supermodules.

Let $\lambda ,\gamma \in \mathfrak {h}^\ast $ and $\Delta (\gamma ) \xrightarrow {f} \Delta (\lambda )$ be a nonzero homomorphism between the corresponding Verma supermodules over $\mathfrak g$ . By Theorem 2, there exists a homomorphism $\tilde {f}\in \text {Hom}_{\mathcal O}(\Delta _{\bar 0}(\gamma ), \Delta _{\bar 0}(\lambda ))$ such that $K(\tilde {f}) =f$ . Applying the Kac functor to the short exact sequence

$$ \begin{align*}0\rightarrow \Delta_{\bar 0}(\gamma) \xrightarrow{\tilde{f}} \Delta_{\bar 0}(\lambda) \rightarrow \Delta_{\bar 0}(\lambda)/\Delta_{\bar 0}(\gamma)\rightarrow 0\end{align*} $$

in $\mathcal O_{\bar 0}$ , we obtain the short exact sequence

$$ \begin{align*}0\rightarrow \Delta(\gamma) \xrightarrow{f} \Delta(\lambda) \rightarrow K(\Delta_{\bar 0}(\lambda)/\Delta_{\bar 0}(\gamma))\rightarrow 0\end{align*} $$

in $\mathcal O$ . This gives rise to the isomorphism

$$ \begin{align*}\text{soc}(\Delta(\lambda)/\Delta(\gamma)) \cong \text{soc}(K(\Delta_{\bar 0}(\lambda)/\Delta_{\bar 0}(\gamma))) ,\end{align*} $$

where we write $\Delta (\lambda )/\Delta (\gamma )$ for $\Delta (\lambda )/f(\Delta (\gamma ))$ .

Also, by Theorem 2, there exists a dominant integral $\mu $ such that $\lambda = x\cdot \mu $ and $\gamma =y\cdot \mu $ , for some $x,y\in W$ , which satisfy $x<y$ with respect to the Bruhat order. Let

$$ \begin{align*}\theta_\mu^{\text{on}}:(\mathcal O_{\bar 0})_0 \rightarrow (\mathcal O_{\bar 0})_\mu\end{align*} $$

denote the translation functor to the $\mu $ -wall, where $(\mathcal O_{\bar 0})_0$ and $(\mathcal O_{\bar 0})_\mu $ denote the principal block and the block containing $L_{\bar 0}(\mu )$ , respectively. For any $ s\in W$ , we set $n_{x, y, z}$ to be the integer determined by

$$ \begin{align*}\theta_\mu^{\text{on}}(\text{soc}(\Delta_{\bar 0}(x'\cdot 0)/\Delta_{\bar 0}(y'\cdot 0)))=\text{soc}(\Delta_{\bar 0}(x\cdot \mu)/\Delta_{\bar 0}(y\cdot \mu))=\bigoplus_{z\in W}L_{\bar 0}(z\cdot \mu)^{\oplus n_{x, y, z}},\end{align*} $$

where $x',y'$ are the shortest representatives in the cosets $xW_\mu $ and $yW_\mu $ , respectively; see [KMM2, Proposition 15].

For $\nu \in \mathfrak {h}^\ast $ , we recall the weight $\nu ^+$ defined in (3-4). We are now in a position to state the main result of this section which describes the socle of the cokernel of homomorphism between Verma supermodules over $\mathfrak g$ .

Theorem 7. We have the following description of the socle of the cokernel of a nonzero homomorphism $\Delta (\gamma ) \rightarrow \Delta (\lambda )$ :

$$ \begin{align*}\mathrm{soc}(\Delta(\lambda)/\Delta(\gamma))\cong \bigoplus_{z\in W}L((z\cdot \mu)^++\eta)^{\oplus n_{x,y,z}}. \end{align*} $$

Proof. By [CM, Lemma 3.2] (see also [CCM, Theorem 51]),

$$ \begin{align*} \begin{array}{rcl} \text{soc}(\Delta(\lambda )/\Delta(\gamma)) &\cong &\text{soc}(K(\Delta_{\bar 0}(\lambda)/\Delta_{\bar 0}(\gamma))) \\[3pt] &= &U(\mathfrak g)\cdot (\Lambda^{\text{top}} \mathfrak g_{-1}\otimes \text{soc}(\Delta_{\bar 0}(\lambda)/\Delta_{\bar 0}(\gamma))) \\[3pt] &\cong& \text{head}\,\operatorname{Ind}\nolimits_{\mathfrak{g}_{\leq 0}}^{\mathfrak{g}} \bigg(\bigoplus\limits_{z\in W} L_{\bar 0}(z\cdot 0 +\eta)^{\oplus n_{x,y,z}}\bigg). \end{array} \end{align*} $$

It follows that

$$ \begin{align*}\mathrm{soc}(\Delta(\lambda)/\Delta(\gamma))\cong \bigoplus\limits_{z\in W}L^{r}(z\cdot 0 +\eta)^{\oplus n_{x,y,z}}= \bigoplus_{z\in W}L((z\cdot \mu)^++\eta)^{\oplus n_{x,y,z}}.\end{align*} $$

This completes the proof.

Remark 8. Let $\lambda $ be a dominant and regular weight. By Theorem 7, for any $y\in W$ , the quotient $\Delta (\lambda )/\Delta (y\cdot \lambda )$ has simple socle if and only if y is a bigrassmannian permutation; see [KMM2, Section 1.1 and Theorem 1.3].

For an integral weight $\lambda \in \mathfrak {h}^\ast $ , we let $\overline \lambda \in W\cdot \lambda $ and $\underline \lambda \in W\cdot \lambda $ be the unique dominant and the unique antidominant element, respectively. Theorem 7 has the following consequence.

Corollary 9. Suppose that $\mu \in \mathfrak {h}^\ast $ is not antidominant. Then we have

$$ \begin{align*}\mathrm{dim\ Ext}_{\mathcal O}^1(L(\mu),\Delta(\lambda)) = [\mathrm{soc}\ (\Delta(\overline \lambda)/\Delta(\lambda)): L(\mu)].\end{align*} $$

Proof. Let

$$ \begin{align*} &0\rightarrow \Delta(\lambda) \rightarrow M \rightarrow L(\mu) \rightarrow 0 \end{align*} $$

be a nonsplit short exact sequence. Because it is nonsplit, the socle of M coincides with the socle $L(\underline \lambda )$ of $ \Delta (\lambda ).$ Therefore, M is a submodule of the injective envelope $I(\underline \lambda ) =T(\overline \lambda )$ . Now, we consider a two step projective–injective copresentation of $\Delta (\overline \lambda )$ (whose existence follows, for example, from [AM, Theorem 7.3]):

$$ \begin{align*}0\rightarrow \Delta(\overline \lambda) \rightarrow T(\overline \lambda) \xrightarrow{f} U. \end{align*} $$

By our assumption, $[M/\text {soc}(M): L(\zeta )]>0$ implies that $\zeta $ is not antidominant. As a consequence, $f(M)=0$ because the socle of the projective–injective module U is a  direct sum of simple modules of antidominant highest weights. This means that M is a submodule of $\Delta (\overline \lambda )$ . Consequently, $L(\mu )\cong M/\Delta (\lambda )$ corresponds to a socle constituent of $\Delta (\overline \lambda )/\Delta (\lambda )$ . This completes the proof.

Example 10. We consider $\mathfrak g= \mathfrak {pe}(2).$ Let $\lambda , \mu \in \mathfrak {h}^\ast $ be integral. Suppose that $\mu $ is not antidominant. Set $\overline \lambda =a \epsilon _2+b\epsilon _2$ . Then, by [ChC, Lemma 6.1], we have

$$ \begin{align*} \text{soc}(\Delta(\overline \lambda) /\Delta(\lambda))= \begin{cases} L(\overline \lambda-\omega_2)\quad& \text{if }a =b. \\ L(\overline \lambda)\quad& \text{if }a>b.\\ 0\quad&\text{if }\lambda =\overline \lambda. \end{cases} \end{align*} $$

It follows that

$$ \begin{align*} &\text{dim}\,\text{Ext}^1_{\mathcal O}(L(\mu), \Delta(\lambda)) = \begin{cases} 1\quad &\text{if either }\mu =\overline \lambda-\omega_2\text{ for }a=b\text{, or }\mu=\overline \lambda\text{ for }a>b;\\ 0\quad &\text{otherwise.} \end{cases} \end{align*} $$

Remark 11. Theorem 7 and Corollary 9 can be generalized to other classical Lie superalgebras of type I which have type A even part, under the condition that $\text {dim}\,\text {Hom}_{\mathcal O}(\Delta (\gamma ), \Delta (\lambda ))\leq 1$ . In particular, this extra condition is satisfied when $\mathfrak {g} = \mathfrak {gl}(m|n),~ \mathfrak {osp}(2|2n)$ ; see [Mu, Theorem 9.3.2].

5 Homological dimensions

Let $\mathcal A$ be an abelian category. Suppose that $\mathcal A$ has enough projective and injective objects. For any $M\in \mathcal A$ , we let ${\text {pd}}_{\mathcal A}M$ denote the projective dimension of M. We define ${\text {gl.dim}}\,\mathcal A$ to be the global dimension of $\mathcal A$ , namely,

$$ \begin{align*}{\text{gl.dim}}\,\mathcal A:=\mathop{\text{sup}}\limits_{M \in\mathcal A} \{{\text{pd}}_{\mathcal A} M\}.\end{align*} $$

The finitistic dimension of $\mathcal A$ is defined as

$$ \begin{align*} \text{fin.dim}\,\mathcal A : =\mathop{\text{sup}}\limits_{M \in \mathcal A} \{{\text{pd}}_{\mathcal A}M\mid {\text{pd}}_{\mathcal A}M<\infty \}. \end{align*} $$

We simply use ${\text {pd}}({}_-)$ and ${\text {gl.dim}}({}_-)$ when the context is clear. We similarly define the injective dimension ${\text {id}}\,M= {\text {id}}_{\mathcal A}M$ of an object $M\in \mathcal A$ .

In this section, we assume that $\mathfrak g$ is an arbitrary classical Lie superalgebra.

5.1 Finitistic dimension of various categories of supermodules

We denote by $\mathfrak g$ -Mod and $\mathfrak g_{\bar 0}$ -Mod the category of all $\mathfrak g$ -modules and the category of all $\mathfrak g_{\bar 0}$ -modules, respectively.

Let $\tilde {\mathcal C}$ and ${\mathcal C}$ be abelian subcategories of $\mathfrak g$ -Mod and $\mathfrak g_{\bar 0}$ -Mod, respectively. Assume that:

1. (i) $\operatorname {Ind}\nolimits , \mathrm {Coind}$ , and $\operatorname {Res}\nolimits $ restrict to well-defined functors between ${\mathcal C} $ and $\tilde {\mathcal C}$ ;

2. (ii) $\mathcal C$ has enough injective and projective modules;

3. (iii) $\text {gl.dim}\,\mathcal C <\infty $ .

The following theorem generalizes [CS, Theorem 4.1] (also see [CS, Lemma 4.3]) where the case of the superalgebra $\mathfrak {gl}(m|n)$ was considered.

Theorem 12. We have fin.dim $ \,\tilde {\mathcal C} = $ gl.dim $\,\mathcal C$ .

Proof. Because $\operatorname {Ind}\nolimits $ is isomorphic to $\mathrm {Coind}$ , up to parity change, it follows that the functors $\operatorname {Ind}\nolimits $ , $\mathrm {Coind}$ , and $\operatorname {Res}\nolimits $ send injective (respectively projective) objects to injective (respectively projective) objects.

The category $\tilde {\mathcal C}$ is a Frobenius extension of $\mathcal C$ in the sense of [Co, Definition 2.2.1]. Therefore, $\tilde {\mathcal C}$ has enough projective and injective objects by [Co, Proposition 2.2.1]. Moreover, every injective (respectively projective) object in $\tilde {\mathcal C}$ is a direct summand of an object induced from an injective (respectively projective) object of $\mathcal C$ .

Consequently, all injective objects in $\tilde {\mathcal C}$ have finite projective dimension. By an argument used in the proof of [Ma3, Theorem 3] (see also [Co, Corollary 3.6.10]), we have that $\text {fin.dim}\, \tilde {\mathcal C}$ is the maximum of the projective dimensions of injective objects in $\tilde {\mathcal C}$ . Note also that the global dimension of $\mathcal C$ is the maximum of the projective dimensions of injective objects in $\mathcal C$ .

We observe that for any injective I in $\mathcal C$ and $Y\in \tilde {C}$ , it follows from the Frobenius reciprocity that

$$ \begin{align*}\operatorname{Ext}\limits^d_{\tilde{\mathcal C}}(\operatorname{Ind}\nolimits I, Y) =\operatorname{Ext}\limits^d_{\mathcal C}(I,\operatorname{Res}\nolimits Y), \end{align*} $$

which implies that $\text {fin.dim} \,\tilde {\mathcal C} \leq \text {gl.dim}\,\mathcal C$ .

Finally, let $I\in \mathcal C$ be injective. Then I is a direct summand of $\operatorname {Res}\nolimits \operatorname {Ind}\nolimits I\cong \Lambda ^{\bullet } \mathfrak g_{\bar 1} \otimes I$ , which implies that

$$ \begin{align*}{\text{pd}}\, I \geq {\text{pd}}\, \operatorname{Ind}\nolimits I \geq {\text{pd}} \,\operatorname{Res}\nolimits \operatorname{Ind}\nolimits I \geq {\text{pd}}\, I. \end{align*} $$

This shows that $\text {fin.dim} \,\tilde {\mathcal C} \geq \text {gl.dim}\,\mathcal C$ and the result follows.

Below we apply Theorem 12 to various examples.

Example 13. Let $\tilde {\mathcal C}= \mathfrak g$ -Mod and ${\mathcal C}= g_{\bar 0}$ -Mod. Condition (i) is obvious and conditions (ii) and (iii) are clear because $U(g_{\bar 0})$ is Noetherian. We have $\text {gl.dim}\,\mathcal C =\text {dim}\,\mathfrak g_{\bar 0}$ . Therefore, by Theorem 12, we have

$$ \begin{align*}\text{fin.dim}~\mathfrak g\text{-Mod} = \dim \mathfrak{g}_{\bar 0}.\end{align*} $$

Example 14. Consider the super analog $\mathcal C(\mathfrak g,\mathfrak {h})$ of the category $\mathcal C(\mathfrak g_{\bar 0},\mathfrak {h})$ of weight $\mathfrak g$ -modules from [Zu], that is, $\mathcal C(\mathfrak g,\mathfrak {h})$ , which is the full subcategory of $\mathfrak g$ -Mod consisting of $\mathfrak g$ -modules that are semisimple over $\mathfrak h$ .

For $\lambda \in \mathfrak {h}^\ast $ , let $I_\lambda $ be the left ideal of $U(\mathfrak g)$ generated by $h-\lambda (h)$ for $h\in \mathfrak h$ . Then the quotient of $U(\mathfrak g)$ by $I_\lambda $ is projective in $\mathcal C(\mathfrak {g},\mathfrak {h})$ . Also, every weight $\mathfrak g$ -module is a quotient of a direct sum of the $I_\lambda $ terms. Therefore, $\mathcal C(\mathfrak g_{\bar 0}, \mathfrak {h})$ has enough projective objects. By [Zu, Lemma 2.2], $\mathcal C(\mathfrak g_{\bar 0},\mathfrak h)$ has enough injective objects as well.

Let $M\in \mathcal C(\mathfrak g_{\bar 0},\mathfrak {h})$ . By [Ku, Corollary 3.1.8], the trivial module ${\mathbb C}$ in $\mathcal C(\mathfrak g_{\bar 0},\mathfrak {h})$ has the following finite projective resolution in $\mathcal C(\mathfrak g_{\bar 0},\mathfrak {h})$ :

$$ \begin{align*}\cdots \rightarrow D_2\rightarrow D_1 \rightarrow D_0 \rightarrow {\mathbb C} \rightarrow 0, \end{align*} $$

where $D_p$ is the induced module $U(\mathfrak g_{\bar 0})\otimes _{U(\mathfrak h)}\Lambda ^p(\mathfrak g_{\bar 0}/\mathfrak h)$ for $p\geq 0$ . Tensoring with this resolution (over $\mathbb {C}$ ), we conclude that every $M\in \mathcal C(\mathfrak g_{\bar 0},\mathfrak h)$ has finite projective dimension.

In particular, we have

$$ \begin{align*} &\text{fin.dim}\, \mathcal C(\mathfrak{g},\mathfrak{h})= \text{fin.dim} \,\mathcal C(\mathfrak{g}_{\bar 0},\mathfrak{h}) = \text{dim}\mathfrak{g}_{\bar 0} -\text{dim}\mathfrak{h}. \end{align*} $$

Example 15. Let $\mathcal O^{\mathfrak {p}}$ be the parabolic BGG category of $\mathfrak g$ -modules in the sense of [Ma3, Section 3]. By [CoM2], we have

(5-1) $$ \begin{align} &\text{fin.dim}\, \mathcal O^{\mathfrak{p}} = {\text{gl.dim}} \,\mathcal O_{\bar 0}^{\mathfrak{p}}=2\ell(w_0) -2\ell(w_0^{\mathfrak{p}}), \end{align} $$

where $w_0$ is the longest element in W and $w_0^{\mathfrak {p}}$ is the longest element in the Weyl group of the Levi subalgebra of $\mathfrak {p}$ .

Example 16. Consider $\mathfrak g=\mathfrak {pe} (n)$ with a reduced parabolic subalgebra $\mathfrak {p}$ . For $\lambda \in \mathfrak {h}^\ast $ , let $\text {Irr}\,\mathcal O_{\lambda }^{\mathfrak {p}}$ denote the set of all highest weights of irreducible modules in the parabolic block $\mathcal O_\lambda ^{\mathfrak {p}}$ . Recall the equivalence relation $\sim $ on $\mathfrak {h}^*$ defined in [ChC, Section 5.2] which is transitively generated by

$$ \begin{align*}\begin{cases}\lambda\sim \lambda \pm2\epsilon_k \quad&\mbox{for }1\le k\le n;\\ \lambda\sim w \cdot \lambda \quad&\mbox{for }w\in W_{[\lambda]}. \end{cases}\end{align*} $$

Here $W_{[\lambda ]}$ denotes the integral Weyl group associated to $\lambda $ . It is generated by all reflections $s_\alpha $ with $\langle \lambda , \alpha \rangle \in {\mathbb Z}.$ It is well known that $W_{[\lambda ]}$ is the Weyl group of a certain semisimple Lie algebra (see, for example, [Hu, Theorem 3.4]). We define $w_0^{\lambda }$ to be the longest element of $W_{[\lambda ]}$ . For $\lambda ,\nu \in \Sigma ^+_{\mathfrak {p}}$ , it follows from [CP, Theorem B] (see also [CP, Theorem 3.2]) that $\mathcal O_\lambda ^{\mathfrak {p}} =\mathcal O_\nu ^{\mathfrak {p}}$ if and only if $\lambda \sim \nu $ . As a consequence, there exists $\nu \in \text {Irr}\,\mathcal O_{\lambda }^{\mathfrak {p}}$ which is dominant and regular.

Let $\lambda \in \mathfrak {h}^\ast $ be a dominant weight such that the parabolic block $\mathcal O^{\mathfrak {p}}_\lambda $ (see Section 6.2 for the precise definition) is nonzero. Let $ (\mathcal O^{\mathfrak {p}}_{{\bar 0}})_{\lambda +X}$ denote the full subcategory of $\mathcal O^{\mathfrak {p}}_{{\bar 0}}$ consisting of all modules whose weights belong to the set $\lambda +X$ . Applying, if necessary, the equivalence established in [CMW, Section 2.3] (see also [ChC, Section  4.3]), we can assume that $W_{[\lambda ]}$ is a parabolic subgroup of W. Then the problem of global dimension of $(\mathcal O^{\mathfrak {p}}_{{\bar 0}})_{\lambda +X}$ reduces to that of an integral block for a semisimple Lie algebra of type $\bf A$ by [CMW, Proposition 2.3]. By [CoM2, Theorem 5.2],

$$ \begin{align*} &\text{gl.dim} (\mathcal O^{\mathfrak{p}}_{{\bar 0}})_{\lambda +X} =2\ell(w_0^{\lambda}) -2\ell(w_0^{\mathfrak{p}}). \end{align*} $$

Take $\nu \in \text {Irr}\,\mathcal O_{\lambda }^{\mathfrak {p}}$ dominant and regular. In this case, ${\text {pd}}\, I_{\bar 0}^{\mathfrak {p}}(\nu ) =2 \ell (w_0^{\lambda }) -2\ell (w_0^{\mathfrak {p}})$ by [CoM2, Theorem 4.1 and Proposition 6.9]. By Theorem 12, thus

$$ \begin{align*} &\text{fin.dim}\, \mathcal O_\lambda^{\mathfrak{p}} =2\ell(w_0^{\lambda}) - 2\ell(w_0^{\mathfrak{p}}). \end{align*} $$

5.2 The parabolic BGG category $\mathcal O^{\mathfrak {p}}$

In this section, we let $\mathfrak g$ be an arbitrary classical Lie superalgebra. We let $\langle {}_-,{}_-\rangle $ denote the usual bilinear form on $\mathfrak h^\ast $ .

In the remaining parts of the present paper, we use the notion of parabolic decomposition as defined in [Ma3, Section 2.4] and consider the reduced parabolic subalgebra $\mathfrak {p}$ from [CCC, Section 1.4]. Namely, we assume that $\mathfrak b\subseteq \mathfrak {p}\subseteq \mathfrak g$ where there is a purely even Levi subalgebra $\mathfrak l\subseteq \mathfrak {p}$ . The even part $\mathfrak p_{\bar 0}$ of $\mathfrak p$ is a parabolic subalgebra of $\mathfrak g_{\bar 0}$ arising from a parabolic decomposition of $\mathfrak g_{\bar 0}$ . In what follows, we always assume that $\mathfrak {p}_{\bar 1} =\mathfrak {g}_1$ in the case when $\mathfrak g$ is of type I.

The parabolic category $\mathcal O^{\mathfrak {p}}$ (respectively $\mathcal O^{\mathfrak {p}_{\bar 0}}_{\bar 0}$ ) is the full subcategory of $\mathcal O$ (respectively $\mathcal O_{\bar 0}$ ) consisting of all finitely generated $\mathfrak g$ -modules (respectively $\mathfrak g_{\bar 0}$ -modules) on which $\mathfrak {p}$ (respectively $\mathfrak {p}_{\bar 0}$ ) acts locally finitely. For simplicity of notation, we define $\mathcal O^{\mathfrak {p}}_{\bar 0}:= \mathcal O^{\mathfrak {p}_{\bar 0}}_{\bar 0}$ , and, in what follows, we omit ${\bar 0}$ from $(-)^{\mathfrak {p}_{\bar 0}}$ or $(-)_{\mathfrak {p}_{\bar 0}}$ when defining notation for $\mathfrak {g}_{\bar 0}$ -modules.

For any $\lambda \in \mathfrak {h}^\ast $ , let $L^{\mathfrak l}(\lambda )$ denote the irreducible $\mathfrak l$ -module with highest weight $\lambda $ . We define the set of $\mathfrak l$ -dominant weights (called $\mathfrak {p}$ -dominant weights in [CCC, Section 3]) as follows:

$$ \begin{align*} &\Sigma^+_{\mathfrak{p}}:=\{\lambda\in \mathfrak h^\ast\mid \langle\lambda, \alpha\rangle \in \mathbb Z_{\geq 0}, \text{ for all } \alpha \in \Phi^+(\mathfrak l) \}. \end{align*} $$

The corresponding parabolic category $\mathcal O^{\mathfrak {p}}$ is the Serre subcategory of $\mathcal O$ generated by $\{L(\lambda ) \, | \, \lambda \in \Sigma _{\mathfrak {p}}^+\}$ .

Let $\mathcal O^{\mathfrak {p}}_\lambda $ denote the (indecomposable) block in $\mathcal O^{\mathfrak {p}}$ containing $L(\lambda )$ . In particular, $\mathcal O_\lambda :=\mathcal O_\lambda ^{\mathfrak b}$ is the (indecomposable) block containing $L(\lambda )$ in category $\mathcal O$ .

For $\lambda \in \Sigma _{\mathfrak {p}}^+$ , we define the parabolic Verma $\mathfrak g_{\bar 0}$ -module and the parabolic Verma $\mathfrak g$ -module, respectively, as

$$ \begin{align*} &\Delta^{\mathfrak{p}}_{\bar 0}(\lambda) : = \operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak g_{\bar 0}}L^{\mathfrak l}(\lambda)\quad\text{and}\quad \Delta^{\mathfrak{p}}(\lambda):= \operatorname{Ind}\nolimits_{\mathfrak p}^{\mathfrak g}L^{\mathfrak l}(\lambda). \end{align*} $$

Similarly, we can define the dual parabolic Verma modules $\nabla ^{\mathfrak {p}}_{\bar 0}(\lambda )$ and $\nabla ^{\mathfrak {p}}(\lambda )$ in $\mathcal O^{\mathfrak {p}}_{\overline 0}$ and $\mathcal O^{\mathfrak {p}}$ , respectively (see [CCC, Definition 3.2]). We have

$$ \begin{align*}\Delta^{\mathfrak{p}}(\lambda) \twoheadrightarrow L(\lambda)\quad\text{and}\quad L(\lambda) \hookrightarrow \nabla^{\mathfrak{p}}(\lambda).\end{align*} $$

Let $P^{\mathfrak {p}}(\lambda )$ be the projective cover of $L(\lambda )$ in $\mathcal O^{\mathfrak {p}}$ and $I^{\mathfrak {p}}(\lambda )$ be the injective envelope of $L(\lambda )$ in $\mathcal O^{\mathfrak {p}}$ . For $\mathcal O^{\mathfrak {p}} =\mathcal O$ , we denote the corresponding objects by $P(\lambda )$ and $I(\lambda )$ . Similarly, we define the indecomposable projectives $P_{\bar 0}(\lambda )$ , $P_{\bar 0}^{\mathfrak {p}}(\lambda )$ and injectives $I_{\bar 0}(\lambda )$ , $I^{\mathfrak {p}}_{\bar 0}(\lambda )$ in $\mathcal O_{\bar 0}$ and $\mathcal O_{\bar 0}^{\mathfrak {p}}$ , respectively. Also, we denote by $T^{\mathfrak {p}}(\lambda )$ and $T_{\bar 0}^{\mathfrak {p}}(\lambda )$ the tilting modules of highest weight $\lambda $ in $\mathcal O^{\mathfrak {p}}$ and $\mathcal O_{\bar 0}^{\mathfrak {p}}$ , respectively. Finally, we note that $\operatorname {Ind}\nolimits : \mathcal O_{\bar 0}^{\mathfrak {p}}\rightarrow \mathcal O^{\mathfrak {p}}$ and $\operatorname {Res}\nolimits : \mathcal O^{\mathfrak {p}}\rightarrow \mathcal O_{\bar 0}^{\mathfrak {p}}$ are well defined.

Also, we note that

$$ \begin{align*}\Delta^{\mathfrak{p}}_\lambda= K(\Delta^{\mathfrak{p}}_{\bar 0}(\lambda)),~\nabla^{\mathfrak{p}}(\lambda):= \operatorname{Ind}\nolimits_{\mathfrak g_{\bar 0}+\mathfrak g _{-1}}^{\mathfrak g}(\nabla^{\mathfrak{p}}_{\bar 0}(\lambda) \otimes \Lambda^{\text{top}}\mathfrak g_1^\ast)\cong \mathrm{Coind}_{\mathfrak p_{\bar 0}+\mathfrak g_{-1}}^{\mathfrak g}L^{\mathfrak l}(\lambda),\end{align*} $$

in the case when $\mathfrak g$ is of type I.

We define $w_0^{\mathfrak {p}}$ to be the longest element in the Weyl group of $\mathfrak l$ . Finally, we set $w_0:=w_{0}^{\mathfrak b}$ .

5.3 Projective dimension of modules in $\mathcal O^{\mathfrak {p}}$

In this subsection, we suppose that $\mathfrak g$ is of type I.

5.3.1 Preliminary results

The following lemma is the parabolic version of [CS, Lemma 4.3].

Lemma 17.

1. (1) For $M\in \mathcal O^{\mathfrak {p}}$ , we have pd $\, M \geq $ pd $\, \operatorname {Res}\nolimits M$ .

2. (2) For $N\in \mathcal O^{\mathfrak {p}}_{\bar 0}$ , we have pd $\, N = $ pd $\, \operatorname {Ind}\nolimits N =$ pd $\, \mathrm {Coind} N$ .

Proof. As already mentioned, $\operatorname {Res}\nolimits $ , $\operatorname {Ind}\nolimits $ , and $\mathrm {Coind}$ send projective resolutions to projective resolutions. Also, N is a direct summand of $\operatorname {Res}\nolimits \operatorname {Ind}\nolimits N$ . This implies all statements.

Corollary 18. For any weight $\zeta \in \Sigma ^+_{\mathfrak {p}},$ we have

$$ \begin{align*}{\mathrm{pd}}\, \Delta^{\mathfrak{p}}(\zeta) \geq {\mathrm{pd}} \, \Delta_{\bar 0}^{\mathfrak{p}}(\zeta) \quad\text{and}\quad {\mathrm{pd}}\, \nabla^{\mathfrak{p}}(\zeta) \geq {\mathrm{pd}} \, \nabla_{\bar 0}^{\mathfrak{p}}(\zeta).\end{align*} $$

Proof. Noting that the modules $\Delta _{\bar 0}^{\mathfrak {p}}(\zeta )$ and $\nabla _{\bar 0}^{\mathfrak {p}}(\zeta )$ are direct summands of $\operatorname {Res}\nolimits \Delta ^{\mathfrak {p}}(\zeta )$ and $\operatorname {Res}\nolimits \nabla ^{\mathfrak {p}}(\zeta )$ , respectively, the statement follows from Lemma 17.

The following result is a consequence of the combination of [Co, Corollary 3.2.5] and [Co, Theorem 8.2.1]. We provide more details in the proof.

Proposition 19. Suppose that $\mathfrak g$ is of type I. Then, for each $\lambda \in \Sigma ^+_{\mathfrak {p}}$ , we have

$$ \begin{align*}{\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}}\, I^{\mathfrak{p}}(\lambda) = {\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}_{\bar 0}}\, I^{\mathfrak{p}}_{\bar 0}(\lambda).\end{align*} $$

Proof. The proof of [CCC, Corollary 4.7] shows that $L_{\bar 0}(\lambda )$ is a quotient of $\operatorname {Res}\nolimits L(\lambda )$ . Therefore,

$$ \begin{align*} &\text{Hom}_{\mathcal O^{\mathfrak{p}}}(L(\lambda), \mathrm{Coind}\, I^{\mathfrak{p}}_{\bar 0} (\lambda)) = \text{Hom}_{\mathcal O_{\bar 0}^{\mathfrak{p}}}(\operatorname{Res}\nolimits L(\lambda), I^{\mathfrak{p}}_{\bar 0} (\lambda)) = [\operatorname{Res}\nolimits L(\lambda): L_{\bar 0}(\lambda)]\neq 0. \end{align*} $$

This means that $L(\lambda ) \hookrightarrow \mathrm {Coind} I_{\bar 0}^{\mathfrak {p}}(\lambda )$ and so $I^{\mathfrak {p}}(\lambda )$ must be a direct summand of $\mathrm {Coind} I_{\bar 0}^{\mathfrak {p}}(\lambda )$ . Consequently ${\text {pd}}\, I^{\mathfrak {p}}(\lambda ) \leq {\text {pd}}\, I_{\bar 0}^{\mathfrak {p}}(\lambda ).$

Next, we claim that $L_{\bar 0}(\lambda )$ is a direct summand of the socle of $\operatorname {Res}\nolimits I^{\mathfrak {p}}(\lambda )$ . We start by observing that

$$ \begin{align*} &\operatorname{Ind}\nolimits L_{\bar 0}(\lambda) =\operatorname{Ind}\nolimits^{\mathfrak{g}}_{\mathfrak g_{\geq 0}} \operatorname{Ind}\nolimits^{\mathfrak{g}_{\geq 0}}_{\mathfrak g_{\bar 0}} L_{\bar 0}(\lambda) \cong \operatorname{Ind}\nolimits^{\mathfrak{g}}_{\mathfrak g_{\geq 0}} (\Lambda^{\bullet} \mathfrak g_{1}\otimes L_{\bar 0}(\lambda)). \end{align*} $$

This implies that

$$ \begin{align*} &\text{Hom}_{\mathcal O^{\mathfrak{p}}_{\bar 0}}(L_{\bar 0}(\lambda), \operatorname{Res}\nolimits I^{\mathfrak{p}} (\lambda)) = \text{Hom}_{\mathcal O^{\mathfrak{p}}}(\operatorname{Ind}\nolimits L_{\bar 0}(\lambda), I^{\mathfrak{p}} (\lambda))\\ &\quad = [\operatorname{Ind}\nolimits L_{\bar 0}(\lambda): L(\lambda)]\geq[\operatorname{Ind}\nolimits^{\mathfrak g}_{\mathfrak g_{\geq 0}} L_{\bar 0}(\lambda): L(\lambda)] \neq 0. \end{align*} $$

Taking Lemma 17 into account, ${\text {pd}}\, I_{\bar 0}^{\mathfrak {p}}(\lambda )\leq {\text {pd}}\, \operatorname {Res}\nolimits I^{\mathfrak {p}}(\lambda ) \leq {\text {pd}} \,I^{\mathfrak {p}}(\lambda )$ . This completes the proof.

The following result is also a consequence of the combination of [Co, Corollary 3.2.5] and [Co, Theorem 8.2.1]. We provide more details in the proof.

Proposition 20. Suppose that $\mathfrak g$ is of type I. Then, for each $\lambda \in \Sigma ^+_{\mathfrak {p}}$ ,

$$ \begin{align*}{\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}}\, T^{\mathfrak{p}}(\lambda) = {\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}_{\bar 0}}\, T^{\mathfrak{p}}_{\bar 0}(\lambda).\end{align*} $$

Proof. Define $\xi _{\pm }\in \mathfrak {h}^\ast $ such that ${\mathbb C}_{\xi _\pm }\cong \Lambda ^{\text {top}} \mathfrak g_{\pm 1}$ , as $\mathfrak g_{\bar 0}$ -modules. We note that

$$ \begin{align*} &\operatorname{Ind}\nolimits \Delta_{\overline 0}^{\mathfrak{p}}(\lambda) \cong \operatorname{Ind}\nolimits_{\mathfrak g_{\overline 0}}^{\mathfrak g} \operatorname{Ind}\nolimits_{\mathfrak p_{\overline 0}}^{\mathfrak g_{\overline 0}}L_{\mathfrak l}(\lambda) \cong \operatorname{Ind}\nolimits_{\mathfrak p}^{\mathfrak g} \operatorname{Ind}\nolimits_{\mathfrak p_{\overline 0}}^{\mathfrak p}L_{\mathfrak l}(\lambda). \end{align*} $$

This means that $\operatorname {Ind}\nolimits \Delta _{\overline 0}^{\mathfrak {p}}(\lambda ) $ has a $\Delta ^{\mathfrak {p}}$ -flag starting at $\Delta ^{\mathfrak {p}}(\lambda +\xi _+)$ . Therefore, the module $T^{\mathfrak {p}}(\lambda +\xi _+)$ is a direct summand of the module $\operatorname {Ind}\nolimits T_{\bar 0}^{\mathfrak {p}}(\lambda )$ . Using Lemma 17, we conclude that

$$ \begin{align*}{\text{pd}}\, T^{\mathfrak{p}}(\lambda) \leq {\text{pd}}\, \operatorname{Ind}\nolimits T^{\mathfrak{p}}_{\bar 0} (\lambda -\xi_+) ={\text{pd}} \,T^{\mathfrak{p}}_{\bar 0}(\lambda-\xi_+)={\text{pd}} \,T^{\mathfrak{p}}_{\bar 0}(\lambda).\end{align*} $$

Here the last equality follows by tensoring with $\Lambda ^{\text {top}} \mathfrak g_{\pm 1}$ .

Next, we claim that $ T_{\bar 0}^{\mathfrak {p}}(\lambda +\xi _-) \hookrightarrow \operatorname {Res}\nolimits T^{\mathfrak {p}}(\lambda )$ . To see this, we note that

$$ \begin{align*} &\Delta^{\mathfrak{p}}_{\bar 0}(\lambda +\xi_-) \hookrightarrow \Lambda^{\bullet} \mathfrak g_{-1}\otimes \Delta^{\mathfrak{p}}_{\bar 0}(\lambda) \cong \operatorname{Res}\nolimits \Delta^{\mathfrak{p}}(\lambda) \hookrightarrow \operatorname{Res}\nolimits T^{\mathfrak{p}}(\lambda). \end{align*} $$

This implies that there is an indecomposable direct summand of $\operatorname {Res}\nolimits T^{\mathfrak {p}}(\lambda )$ that has a $\Delta ^{\mathfrak {p}}_{\bar 0}$ -flag starting at $\Delta ^{\mathfrak {p}}_{\bar 0}(\lambda +\xi _-)$ . In particular, $T_{\bar 0}^{\mathfrak {p}}(\lambda +\xi _-)$ is a direct summand of $\operatorname {Res}\nolimits T^{\mathfrak {p}}(\lambda )$ . Consequently,

$$ \begin{align*}{\text{pd}} \, T_{\bar 0}^{\mathfrak{p}}(\lambda)= {\text{pd}}\, T_{\bar 0}^{\mathfrak{p}}(\lambda+\xi_-) \leq {\text{pd}}\, \operatorname{Res}\nolimits T^{\mathfrak{p}}(\lambda)\leq {\text{pd}}\, T^{\mathfrak{p}}(\lambda)\end{align*} $$

by Lemma 17. The result of the proposition follows.

5.3.2 Parabolic dimension shift

This subsection is devoted to the following formula for the projective dimension.

Theorem 21. Suppose that $M\in \mathcal O^{\mathfrak {p}}$ with ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}M<\infty $ . Then,

$$ \begin{align*} &{\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}}M = {\mathrm{pd}}_{\mathcal O}M -2\ell(w_0^{\mathfrak{p}}). \end{align*} $$

Proof. We first claim that

$$ \begin{align*}{\text{pd}}_{\mathcal O^{\mathfrak{p}}}M < \infty \Rightarrow {\text{pd}}_{\mathcal O}M < \infty. \end{align*} $$

To see this, we let Q be a projective module in $\mathcal O^{\mathfrak {p}}$ . The adjoint pair $\operatorname {Ind}\nolimits $ and $\operatorname {Res}\nolimits $ of functors between $\mathcal O_{\bar 0}^{\mathfrak {p}}$ and $\mathcal O^{\mathfrak {p}}$ gives an epimorphism $\operatorname {Ind}\nolimits \operatorname {Res}\nolimits Q \rightarrow Q$ , and so Q is a direct summand of $\operatorname {Ind}\nolimits \operatorname {Res}\nolimits Q$ . Using Frobenius reciprocity, it follows that

$$ \begin{align*}\text{Ext}_{\mathcal O}^d(\operatorname{Ind}\nolimits \operatorname{Res}\nolimits Q, N)\cong \text{Ext}_{\mathcal O_{\bar 0}}^d( \operatorname{Res}\nolimits Q, \operatorname{Res}\nolimits N)=0,\end{align*} $$

for any $d>{\text {pd}}_{\mathcal O_{\bar 0}}\operatorname {Res}\nolimits Q$ and $N\in \mathcal O$ . As a consequence, ${\text {pd}}_{\mathcal O}Q<\infty .$

Now suppose that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}M < \infty $ . This means that M has a finite projective resolution in $\mathcal O^{\mathfrak {p}}$ :

$$ \begin{align*} 0 \rightarrow Q^d\xrightarrow{f^d} Q^{d-1}\xrightarrow{f^{d-1}} \cdots \xrightarrow{f^2} Q^1\xrightarrow{f^1} Q^0 \xrightarrow{f^0} M \rightarrow 0.\end{align*} $$

We define $K^s$ to be the kernel of the map $f^s$ for each s. Then the inequality ${\text {pd}}_{\mathcal O}K^{s}<\infty $ together with the exactness of $0\rightarrow K^s \rightarrow Q^{s-1} \rightarrow K^{s-1}\rightarrow 0$ imply that ${\text {pd}}_{\mathcal O}K^{s-1}<\infty $ . Consequently, we have ${\text {pd}}_{\mathcal O}M < \infty $ .

Now suppose that both ${\text {pd}}_{\mathcal O}M$ and ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}M$ are finite. Then, by [Co, Corollary 3.2.5] and [Co, Theorem 7.2.1(iii)],

$$ \begin{align*}{\text{pd}}_{\mathcal O}M={\text{pd}}_{\mathcal O_{\bar 0}}\operatorname{Res}\nolimits M \quad\text{and}\quad{\text{pd}}_{\mathcal O^p}M={\text{pd}}_{\mathcal O^{\mathfrak{p}}_0}\operatorname{Res}\nolimits M.\end{align*} $$

The proof now follows from [CoM2, Theorem 4.1].

Corollary 22. Let $M \in \mathcal O^{\mathfrak {p}}$ with ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}} M<\infty .$

1. (1) M is projective in $\mathcal O^{\mathfrak {p}}$ if and only if ${\mathrm {pd}}_{\mathcal O}M =2\ell (w_0^{\mathfrak {p}})$ .

2. (2) If $\mathfrak {p}=\mathfrak {g}$ , then ${\mathrm {pd}}_{\mathcal O}M =2\ell (w_0)$ .

Proof. Claim $(1)$ follows directly from Theorem 21.

If $\mathfrak {p}=\mathfrak {g}$ , then $\mathcal O^{\mathfrak {p}}$ is the category of finite dimensional modules. In this case, we have $\text {fin.dim}\,\mathcal O^{\mathfrak {p}} = 2\ell (w_0) - 2\ell (w_0^{\mathfrak {p}}) =0$ by (5-1), which implies that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}M=0$ . Claim $(2)$ now follows from Theorem 21.

5.4 Injective dimension of modules in $\mathcal O^{\mathfrak {p}}$

In this subsection, unless mentioned otherwise, we suppose that $\mathfrak {g}$ is of type I.

Let $\mu $ be a dominant integral weight whose singularity determines the parabolic subalgebra ${\mathfrak {p}}_{\bar 0}$ , namely, the Levi subalgebra of ${\mathfrak {p}}_{\bar 0}$ is generated by $\mathfrak h$ and ${\mathfrak {g}}_\alpha $ with roots satisfying $\langle \mu +\rho _{\bar 0}, \alpha \rangle =0$ . Following [CCC, Section 1.4], there is a parabolic subalgebra $\hat {\mathfrak {p}}$ (in the sense of [Ma3, Section 2.4]) of $\mathfrak g$ such that $\hat {\mathfrak {p}}_{\bar 0}$ is determined by $\hat \mu := -w_0\mu $ . For instance, if $\mathfrak g$ is of type I, then $\hat {\mathfrak {p}}= \hat {\mathfrak {p}}_{\bar 0} \oplus {\mathfrak g}_{-1}$ . Also, we have $\hat {\mathfrak b} ={\mathfrak b}^r$ for $\mathfrak {pe} (n)$ from Section 3.2.

We recall, see [CCC, Section 1.3], that there is a duality (that is, an antiequivalence) $\bf D: \mathcal O^{\mathfrak {p}}\rightarrow \mathcal O^{\hat {\mathfrak {p}}}$ such that

(5-2) $$ \begin{align} &\mathbf{D} L^{\mathfrak{p}}(\lambda) =L^{\hat {\mathfrak{p}}}(-w_0\lambda), ~\mathbf{D}\Delta^{\mathfrak{p}}(\lambda) = \nabla^{\hat {\mathfrak{p}}}(-w_0\lambda), \text{ and }\mathbf{D}T^{\mathfrak{p}}(\lambda) = T^{\hat {\mathfrak{p}}}(-w_0\lambda). \end{align} $$

We refer to [CCC, Proposition 3.4 and Lemma 3.6] for more details. Applying the results from Section 5.3, we obtain the following proposition.

Proposition 23. For any $M\in \mathcal O^{\mathfrak {p}}$ , we have $ {\mathrm {id}}_{\mathcal O^{\mathfrak {p}}}M= {\mathrm {pd}}_{\mathcal O^{\hat {\mathfrak {p}}}}\mathbf {D}M.$

Furthermore, we have the following properties.

1. (1) Suppose that ${\mathrm {pd}}_{\mathcal O^{ \mathfrak {p}}} M<\infty $ , then

   (5-3) $$ \begin{align} &{\mathrm{id}}_{\mathcal O^{\mathfrak{p}}} M = {\mathrm{id}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} \operatorname{Res}\nolimits M. \end{align} $$

2. (2) Suppose that ${\mathrm {id}}_{\mathcal O^{ \mathfrak {p}}} M<\infty $ , then

   (5-4) $$ \begin{align} &{\mathrm{id}}_{\mathcal O^{\mathfrak{p}}}M = {\mathrm{id}}_{\mathcal O}M -2\ell(w_0^{\mathfrak{p}}). \end{align} $$

3. (3) Suppose that $\mathfrak {g}$ is of type I. Then, for any $\lambda \in \Sigma ^+_{\mathfrak {p}}$ ,

   $$ \begin{align*} &{\mathrm{id}}_{\mathcal O^{\mathfrak{p}}}\, P^{\mathfrak{p}}(\lambda) ={\mathrm{id}}_{\mathcal O^{\mathfrak{p}}_{\bar 0}}\, P_{\bar 0}^{ \mathfrak{p}}(\lambda),\\ &{\mathrm{id}}_{\mathcal O^{\mathfrak{p}}}\, T^{\mathfrak{p}}(\lambda) = {\mathrm{id}}_{\mathcal O^{\mathfrak{p}}_{\bar 0}} \,T_{\bar 0}^{ \mathfrak{p}}(\lambda). \end{align*} $$

Proof. Applying $\bf D$ ,

$$ \begin{align*} &{\text{id}}_{\mathcal O^{\mathfrak{p}}} M ={\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}}\mathbf{D}M. \end{align*} $$

Suppose that ${\text {pd}}_{\mathcal O^{\hat {\mathfrak {p}}}}\mathbf {D}M <\infty $ . We note that $\operatorname {Res}\nolimits $ intertwines $\bf D$ and the usual duality on $\mathcal O^{\mathfrak {p}}_{\bar 0}$ which we also denote by $\bf D$ , abusing notation (see the proof of [CCC, Theorem 3.7]). Therefore, from [Co, Corollary 3.2.5] and [Co, Theorem 7.2.1], it follows that

$$ \begin{align*}{\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}}\mathbf{D}M= {\text{pd}}_{\mathcal O_{\bar 0}^{\hat {\mathfrak{p}}}}\operatorname{Res}\nolimits\mathbf{D}M={\text{pd}}_{\mathcal O_{\bar 0}^{\hat {\mathfrak{p}}}}\mathbf{D}\operatorname{Res}\nolimits M={\text{id}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} \operatorname{Res}\nolimits M.\end{align*} $$

This proves part $(1)$ and (5-3).

To prove part $(2)$ , we note that from Theorem 21 and the proof of [Ma3, Theorem  3], we obtain ${\text {id}}_{\mathcal O^{\mathfrak {p}}} M<\infty \Rightarrow {\text {id}}_{\mathcal O}M <\infty $ . Therefore, we may assume that both ${\text {id}}_{\mathcal O} M$ and ${\text {id}}_{\mathcal O^{\mathfrak {p}}}M$ are finite. From (5-3) and [CoM2, Theorem 4.1],

$$ \begin{align*} &{\text{id}}_{\mathcal O^{\mathfrak{p}}} M = {\text{id}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} \operatorname{Res}\nolimits M = {\text{id}}_{\mathcal O_{\bar 0}} \operatorname{Res}\nolimits M-2\ell(w_0^{\mathfrak{p}}) ={\text{id}}_{\mathcal O} M -2\ell(w_0^{\mathfrak{p}}). \end{align*} $$

This proves (5-4) and part $(2)$ . An alternative proof of part $(2)$ arises from the following observation:

$$ \begin{align*} {\text{pd}}_{\mathcal O^{\mathfrak{p}}}M = {\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}}\mathbf{D}M &= {\text{pd}}_{\mathcal O}\mathbf{D}M -2\ell(w_0^{\hat {\mathfrak{p}}}) \\ &= {\text{pd}}_{\mathcal O}\mathbf{D}M -2\ell(w_0^{\mathfrak{p}}) = {\text{pd}}_{\mathcal O^{\mathfrak{p}}}\mathbf{D}M={\text{id}}_{\mathcal O^{\mathfrak{p}}}M. \end{align*} $$

It remains to prove part $(3)$ . By (5-2),

$$ \begin{align*}{\text{id}}_{\mathcal O^{\mathfrak{p}}} P^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}} I^{\hat {\mathfrak{p}}}(-w_0\lambda),~ {\text{id}}_{\mathcal O^{\mathfrak{p}}} T^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}} T^{\hat {\mathfrak{p}}}(-w_0\lambda).\end{align*} $$

It follows that

$$ \begin{align*}{\text{id}}_{\mathcal O^{\mathfrak{p}}} P^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O_{\bar 0}^{\hat {\mathfrak{p}}}} I_{\bar 0}^{\hat {\mathfrak{p}}}(-w_0\lambda) ={\text{pd}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} I_{\bar 0}^{ \mathfrak{p}}(\lambda)={\text{id}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} P_{\bar 0}^{ \mathfrak{p}}(\lambda), \end{align*} $$

$$ \begin{align*} {\text{id}}_{\mathcal O^{\mathfrak{p}}} T^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O_{\bar 0}^{\hat {\mathfrak{p}}}} T^{\hat {\mathfrak{p}}}_{\bar 0}(-w_0\lambda)= {\text{pd}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} T_{\bar 0}^{ \mathfrak{p}}(\lambda), \end{align*} $$

by Proposition 19, Theorem 20, and [So, Theorem 11].

5.5 A sufficient condition for infiniteness of projective dimension

We continue to suppose that $\mathfrak g$ is of type I. We recall the definition of the associated variety from [DS]. Let $M\in \mathfrak {g}$ -Mod and $x\in \mathfrak g_{\overline 1}$ with $[x,x]=0$ . Define

$$ \begin{align*}M_x:= \text{ker}_x/xM,\end{align*} $$

where $\text {ker}_x$ is the kernel of the map $x: M\rightarrow M$ sending m to $xm$ , where $m\in M.$ Then the associated variety $X_M$ of M is defined as

$$ \begin{align*}X_M:=\{x\in \mathfrak{g}_{\bar 1}\mid [x,x]=0 \text{ and } M_x\neq 0\}.\end{align*} $$

The following statement is obtained [DS, Lemma 2.2(i)].

Lemma 24. Let $M\in \mathfrak {g}\mathrm {-Mod}$ be a summand of $\operatorname {Ind}\nolimits N$ for some $N\in \mathfrak g_{\bar 0}\mathrm {-Mod}$ . Then $X_M=0$ .

Lemma 25. Let $0\rightarrow A \rightarrow B \xrightarrow {f} C \rightarrow 0$ be a short exact sequence in $\mathfrak {g}\mathrm {-Mod}$ . Let $x\in \mathfrak g_{\bar 1}$ be nonzero with $[x,x]=0$ . Suppose that $A_x =B_x =0$ . Then $C_x =0.$

In particular, if $X_A=X_B=0$ , then $X_C=0$ .

Proof. Let $c\in C$ be such that $xc=0$ . We want to show that there exists $b\in B$ such that $c=xf(b)$ , which implies that $C_x =0$ . To see this, we first pick $d\in B$ such that $f(d)=c$ . Then $f(xd)=0$ implies that $xd\in A$ .

Now $x(xd) =x^2d =0$ (owing to $x^2=0$ ) implies $xd\in xA$ and so there is $a\in A$ such that $xd =xa$ because $A_x =0$ . Finally, we note that $x(d-a)=0$ implies that $d-a \in xB$ because $B_x=0$ . Therefore, there is $b\in B$ such that $d-a =xb$ . We now apply f and obtain

$$ \begin{align*}c=f(d)=f(d-a)=xf(b),\end{align*} $$

as desired.

The following observation is known (see, for example, [CS, Section 4 and proof of Theorem 4.1] and [DS, Lemma 2.2]).

Corollary 26. Suppose $\tilde {\mathcal C}$ and $\mathcal C$ are respective subcategories of $\mathfrak {g}$ -Mod and $\mathfrak {g}_{\bar 0}$ -Mod satisfying assumptions (i)–(iii). If $M \in \tilde {\mathcal C}$ is such that $X_M\neq 0$ , then ${\mathrm {pd}}_{\tilde {\mathcal C}} M=\infty $ .

In particular, if $M \in \mathcal O^{\mathfrak {p}}$ is such that $X_M\neq 0$ , then ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}} M=\infty $ .

Proof. Suppose that there exists a finite projective resolution of M in $\tilde {\mathcal C}$ :

$$ \begin{align*}0 \xrightarrow{f^{d+1}} P^d \xrightarrow{f^d} \cdots \xrightarrow{} P^2\xrightarrow{f^2} P^1\xrightarrow{f^1} P^0 \rightarrow M \xrightarrow{f^0} 0.\end{align*} $$

Let $0\leq s\leq d$ . By [Co, Proposition 2.2.1], every projective module $P^s$ is a direct summand of a module induced from $\mathcal C$ to $\tilde {\mathcal C}$ . By Lemma 24, we thus have $X_{P^s} = 0$ .

Define $K^s$ to be the kernel of $f^s$ . For any s, we have an exact sequence

$$ \begin{align*}0\rightarrow K^s \rightarrow P^{s-1} \rightarrow K^{s-1}\rightarrow 0.\end{align*} $$

In particular, the fact that $K^d=P^d$ implies that $K^{d-1}_x=0$ by Lemma 25. By induction on s, we may conclude that $M_x=K^0_x =0$ . The result follows.

Example 27. If $\mathfrak g_{\overline 1}$ contains a nonzero x such that $[x,x]=0$ , then it is obvious (see also [DS, Lemma 2.2]) that the associated variety $X_{\mathbb C}$ of the trivial module ${\mathbb C}$ is nonzero. Therefore, we may conclude that both ${\text {pd}}_{\mathfrak {g}\text {-Mod}}{\mathbb C}$ and ${\text {pd}}_{\mathcal O}{\mathbb C}$ are infinite.

Remark 28. It is worth pointing out that the condition $X_M=0$ is independent of the choice of the parabolic subalgebra $\mathfrak {p}$ (compare with Theorem 21). It is natural to ask the following question.

Question. Let $M\in \mathcal O$ . Is it true that ${\text {pd}}_{\mathcal O} M<\infty \Leftrightarrow X_M=0?$

This question has the affirmative answer for the general linear Lie superalgebra $\mathfrak {gl}(m|n)$ by [CS, Theorem 4.1].

The following corollary addresses infiniteness of projective dimension for $\mathfrak {p}$ -highest weight modules.

Corollary 29. Let $\mathfrak g$ be a Lie superalgebra of type I. Suppose that there is a weight $\alpha $ of $\mathfrak g_1$ such that $-\alpha $ is not a weight of $\Lambda ^{\bullet } \mathfrak g_{-1}\otimes U(\mathfrak n^-_{\bar 0}).$ Then the associated variety of every nonzero quotient of $\Delta ^{\mathfrak {p}}(\lambda )$ is nonzero. Consequently, every such quotient has infinite projective dimension in both $\mathcal O$ and $\mathcal O^{\mathfrak {p}}$ , for any $\lambda \in \Sigma _{\mathfrak {p}}^+$ .

In particular, every block of $\mathcal O^{\mathfrak {p}}$ contains infinitely many simple objects.

Proof. Assume that M is a nonzero quotient of $\Delta ^{\mathfrak {p}}(\lambda )$ of finite projective dimension. Let $X_{\alpha } \in \mathfrak {g}_1$ be a root vector of root $\alpha $ .

Pick a nonzero highest weight vector $v\in M$ . By Corollary 26, $M_{X_{\alpha }}=0$ . Therefore, $X_{\alpha } v=0$ implies that there is $m\in M$ such that $v=X_{\alpha }m$ . However, $\lambda -\alpha $ is not a weight of M, a contradiction.

Suppose that $\mathcal O^{\mathfrak {p}}_\lambda $ is a block of $\mathcal O^{\mathfrak {p}}$ containing only finitely many simples. Combining the existence of tilting modules [Ma3, Section 4.3] (also, see [CCC, Theorem 3.5]), the Ringel duality [Ma3, Section 5.4] (also, see [CCC, Corollary 3.8]), and the BGG reciprocity [CCC, Lemma 3.3] for $\mathcal O^{\mathfrak {p}}$ , we may conclude that $\mathcal O^{\mathfrak {p}}_\lambda $ contains a simple tilting module. This is a contradiction to Proposition 20. This completes the proof.

Example 30. Let $\mathfrak k$ be classical of type I such that there exists $d\in \mathcal Z(\mathfrak k)$ and d acts on $\mathfrak k_{1}$ as a nonzero scalar. Set $\mathfrak g:=\mathfrak k_{\bar 0}\oplus \mathfrak k_{1}$ . Then every simple module in the category $\mathcal O$ for $\mathfrak g$ has infinite projective dimension. For example, let $\mathfrak k = \mathfrak {gl}(m|n)$ . Then d is the unique central element (up to a scalar) of $\mathfrak k_{\bar 0}$ . It follows that every simple module in $\mathcal O$ over $\mathfrak {g}= \mathfrak {gl}(m|n)_{\bar 0}\oplus \mathfrak {gl}(m|n)_1$ has infinite projective dimension.

We are going to apply Corollary 29 to the periplectic Lie superalgebra $\mathfrak {pe} (n)$ later in Proposition 36.

6 Examples: projective dimensions of structural supermodules over $\mathfrak {pe} (n)$ and $\mathfrak {osp}(2|2n)$

6.1 Example I: projective dimension of modules for $\mathfrak {osp}(2|2n)$

The orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$ is the following subsuperalgebra of $\mathfrak {gl}(2|2n)$ :

(6-1) $$ \begin{align} \mathfrak{osp}(2\vert 2n)= \left\{ \left( \begin{array}{cccc} c &0 & x &y\\ 0 & -c& v & u\\ -u^t& -y^t & a &b\\ v^t &x^t & c& -a^t \\ \end{array} \right): \begin{array}{c} c\in {\mathbb C};\,\, x,y,v,u\in {\mathbb C}^{n};\\ a,b,c\in {\mathbb C}^{n^2};\\ b=b^t,\,\, c=c^t. \end{array} \right\}. \end{align} $$

The even part $\mathfrak {osp}(2|2n)_{\bar 0}$ of $\mathfrak {osp}(2|2n)$ is isomorphic to ${\mathbb C} \oplus \mathfrak {sp}(2n)$ and the odd part is isomorphic to ${\mathbb C}^2\otimes {\mathbb C}^{2n}$ . We refer the reader to [CW1, Section 1.1.3] and [Mu, Section  2.3] for more details. Recall from Section 3.2 that we denote by $E_{ab}$ the elementary matrices. We now define

$$ \begin{align*}H_\epsilon:=E_{11}-E_{22},~H_i:=E_{2+i,2+i}-E_{2+i+n,2+i+n}, \quad \text{ for }1\leq i\leq n. \end{align*} $$

The Cartan subalgebra $\mathfrak h$ of $\mathfrak {osp}(2|2n)$ is spanned by $H_\epsilon $ and $H_1,H_2,\ldots , H_n$ . We let $\{\epsilon ,\delta _1,\delta _2,\ldots ,\delta _n\}$ be the basis of $\mathfrak h^\ast $ dual to $\{H_\epsilon ,H_1,H_2,\ldots , H_n\}$ . The bilinear form $({}_-,{}_-):\mathfrak h^\ast \times \mathfrak h^\ast \rightarrow {\mathbb C}$ is given by $(\epsilon ,\epsilon ) =1$ and $(\delta _i,\delta _j) =-\delta _{i,j}$ , for $1\leq i,j\leq n.$

We define the set $\Phi ^+_{\bar 0}$ of even positive roots and the set $\Phi ^+_{\bar 1}$ of odd positive roots by

$$ \begin{align*} &\Phi^+_{\bar 0}:=\{\delta_i \pm\delta_j, 2\delta_p\mid 1\leq i< j <n, ~1\leq p \leq n \},\nonumber\\ &\Phi_{\bar 1}^+:=\{\epsilon \pm \delta_p\mid 1\leq p\leq n\}. \end{align*} $$

This gives rise to a triangular decomposition $\mathfrak g=\mathfrak n^+\oplus \mathfrak h \oplus \mathfrak n^-$ with Borel subalgebra $\mathfrak b:= \mathfrak b_{\bar 0} \oplus \mathfrak g_1.$

6.1.1 Simple supermodules over $\mathfrak {osp}(2|2n)$

The aim of this subsection is to study the projective dimension of simple supermodules over arbitrary basic classical Lie superalgebras of type I including $\mathfrak {osp}(2|2n)$ .

We recall the notation of typical weights for basic classical Lie superalgebras of type I; see, for example, [Gor2, Section 2.4]. Let $X^{-}\in \Lambda ^{\max }\mathfrak g_{-1}$ and $\quad X^{+}\in \Lambda ^{\max }\mathfrak g_{1}$ be nonzero vectors. Then by [CM, Lemma 6.5] (also, see [Gor1, Section 4]) it follows that

$$ \begin{align*} &X^{+} X^- = \Omega + \sum_i x_i r_i y_i, \end{align*} $$

for some $x_i \in \Lambda (\mathfrak g_{ -1})\backslash \mathbb {C}$ , $y_i \in \Lambda (\mathfrak g_{1})\backslash \mathbb {C}$ , $r_i\in U(\mathfrak g_{{\overline 0}})$ , and $\Omega \in Z(\mathfrak g_{\overline 0})$ . A weight $\lambda \in \mathfrak {h}^\ast $ is typical if its evaluation at the Harish-Chandra projection of $\Omega $ is nonzero. It was shown in [Ka2, Theorem 1] that the typicality of $\lambda $ is equivalent to the simplicity of the Kac module of highest weight $\lambda $ . A more general definition of typical Kac modules can be found in [CM, Section 6.2].

The following proposition shows that all simple supermodules of atypical blocks over basic classical Lie superalgebras have infinite projective dimension in $\mathcal O$ . This generalizes [CS, Proposition 5.14] where the case of the superalgebra $\mathfrak {gl}(m|n)$ was considered.

Proposition 31. Let $\mathfrak g$ be a basic classical Lie superalgebra of type I. Suppose that $\lambda $ is atypical. Then the associated variety $X_{L(\lambda )}$ is nonzero. In particular, if $L(\lambda ) \in \mathcal O^{\mathfrak {p}}$ for some $\mathfrak {p}$ , then ${\textrm {pd}}_{\mathcal O^{\mathfrak {p}}}L(\lambda )=\infty .$

Before we prove Proposition 31, we need to recall the notion of odd reflection for basic classical Lie superalgebras; see, for example, [PS1, Lemma 1], [CW1, Section 1.4] and [Mu, Section 3] for more details.

Because $\mathfrak g$ is contragradient, the set of roots of $\mathfrak g_{-1}$ can be obtained from the set of roots of $\mathfrak g_{1}$ by multiplying with the scalar $-1$ . Set $d:=\text {dim}\mathfrak g_1=\text {dim}\mathfrak {g}_{-1}$ . By [CW1, Lemma 1.30 and Proposition 1.32] (also, see [Mu, Theorem 3.13]), the positive root system given by the Borel subalgebra $\mathfrak b_{\bar 0}\oplus \mathfrak g_{-1}$ can be obtained from that of $\mathfrak b$ via a sequence of odd reflections. Namely, we can fix an ordering of positive odd roots

(6-2) $$ \begin{align} &\{\alpha_1, \alpha_2, \ldots, \alpha_d\} = \Phi^+_{\bar 1} \end{align} $$

such that the following conditions are satisfied.

1. (1) For any $1\leq i\leq d$ , the set $R^{i} =\Phi ^+_{\bar 0} \cup \{-\alpha _1,\ldots ,-\alpha _{i}\} \cup \{\alpha _{i+1},\ldots , \alpha _d\}$ forms a positive root system.

2. (2) We set $R^{-1}:=R^0$ . Then $\alpha _i$ is a simple root in $R^{i-1}$ and $-\alpha _i$ is a simple root in $R_{i}$ , for any $1\leq i\leq d$ .

Let us fix root vectors $X_i\in \mathfrak g_1\cap \mathfrak g^{\alpha _i}$ and $Y_i \in \mathfrak g_{-1}\cap \mathfrak g^{-\alpha _i}$ for $1\leq i \leq n$ . Also, let $\mathfrak b^{(i)}$ denote the Borel subalgebra corresponding to $R^i$ , namely, $\mathfrak b^{(i)}_{\bar 0} = \mathfrak b_{\bar 0}$ and $\mathfrak b^{(i)}_{\bar 1}$ is generated by all root vectors of odd roots in $R^i$ . By [CW1, Lemma 1.40], we have the following lemma.

Lemma 32. Suppose that $L\in \mathcal O$ is a simple supermodule having $\mathfrak b^{(i)}$ -highest weight $\mu \in \mathfrak {h}^\ast $ . Let $v\in L$ be a $\mathfrak b^{(i)}$ -highest weight vector.

1. (1) If $Y_{i+1} v=0$ , then v is a $\mathfrak b^{(i+1)}$ -highest weight vector of $L.$

2. (2) If $Y_{i+1} v\neq 0$ , then $Y_{i+1} v$ is a $\mathfrak b^{(i+1)}$ -highest weight vector of L.

Example 33. Let $\mathfrak g= \mathfrak {osp}(2|2n)$ . We start with the simple system (see [CW1, Section 1.3.4])

$$ \begin{align*}\{\epsilon-\delta_1,~\delta_2-\delta_3,\ldots,\delta_{n-1}-\delta_n,~ 2\delta_n\}.\end{align*} $$

Then, by a direct computation using [CW1, Equation (1.44) of Lemma 1.30], the ordering (6-2) in this case can be chosen as follows:

$$ \begin{align*} &\epsilon -\delta_1,~\epsilon-\delta_2,\ldots, \epsilon- \delta_n,~\epsilon +\delta_n,~\epsilon +\delta_{n-1},\ldots,\epsilon+\delta_1. \end{align*} $$

Now we are in a position to prove Proposition 31.

Proof of Proposition 31. Let $v\in L(\lambda )$ and $v'\in K(\lambda )$ be highest weight vectors. Because $\lambda $ is atypical, the kernel of the natural epimorphism $\phi : K(\lambda )\rightarrow L(\lambda )$ is nonzero. By [CM, Lemma 3.2], the kernel of $\phi $ contains $Y_dY_{d-1}\cdots Y_1v'$ and so $Y_dY_{d-1}\cdots Y_1v= 0$ . This means that there is $0\leq \ell < d$ such that $Y_{\ell } Y_{\ell - 1}\cdots Y_1 v\neq 0$ and $Y_{\ell +1} Y_{\ell }\cdots Y_1 v =0$ . By Lemma 32, $Y_{\ell } Y_{\ell - 1}\cdots Y_1 v$ is a highest weight vector of L with respect to both $\mathfrak b^{\ell }$ and $\mathfrak b^{\ell +1}$ . Therefore, the weight spaces corresponding to the weights $\lambda -\alpha _1 +\cdots -\alpha _{\ell }\pm \alpha _{\ell +1}$ of $L(\lambda )$ are zero. In particular, the vector $Y_{\ell } Y_{\ell - 1}\cdots Y_1 v\not \in Y_{\ell +1} L(\lambda )$ . This means that $L(\lambda )_{Y_{\ell +1}}\neq 0$ and the result follows from Corollary 26.

6.1.2 (Dual) Verma supermodules over $\mathfrak {osp}(2|2n)$

In this subsection, we assume that $\mathfrak g=\mathfrak {osp}(2|2n)$ for some positive integer n. The following proposition shows that atypical Verma supermodules have infinite projective dimension. We recall that $\Sigma ^+_{\mathfrak {p}}$ denotes the set of highest weights of simple modules in $\mathcal O^{\mathfrak {p}}$ .

Proposition 34. Suppose that $\lambda \in \Sigma _{\mathfrak {p}}^+$ . Then the following conditions are equivalent:

1. (1) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}\Delta ^{\mathfrak {p}}(\lambda ) <\infty $ ;

2. (2) ${\mathrm {pd}}_{\mathcal O}\Delta ^{\mathfrak {p}}(\lambda ) <\infty $ ;

3. (3) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}\Delta ^{\mathfrak {p}}(\lambda ) = {\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}_{\bar 0}}\Delta ^{\mathfrak {p}}(\lambda ) = {\mathrm {pd}}_{\mathcal O}\Delta ^{\mathfrak {p}}(\lambda ) -2\ell (w_0^{\mathfrak {p}})$ ;

4. (4) $\lambda $ is typical.

Proof. We first note that the implication $(1)\Rightarrow (3)$ follows from [Co, Corollary 3.2.5] and [Co, Lemma 6.3.2]. Therefore, we have $(1)\Leftrightarrow (3)$ . From [Gor2, Theorems 1.3.1 and 1.4.1], it follows that $(4)\Rightarrow (1), (2)$ . Therefore, it remains to show that if $\lambda $ is atypical, then the associated variety of $\Delta ^{\mathfrak {p}}(\lambda )$ is nonzero.

We first define an ordering of roots of $\mathfrak g_{-1}$ (that is, roots in $\Phi _{\bar 1} \backslash \Phi ^+_{\bar 1}$ ) as follows:

$$ \begin{align*} \alpha_1:&=-\epsilon -\delta_1<\alpha_2:=-\epsilon-\delta_2<\cdots< \alpha_{n}:=-\epsilon- \delta_n \end{align*} $$

$$ \begin{align*}&< \alpha_{n+1}:=-\epsilon +\delta_n <\alpha_{n+2}:=-\epsilon +\delta_{n-1}<\cdots<\alpha_{2n}:=-\epsilon+\delta_1. \end{align*} $$

By (6-1), for each $1\leq i \leq n$ , there exist nonzero vectors $X_i \in \mathfrak {g}^{-\alpha _i}$ and $Y_{i}\in \mathfrak {g}^{\alpha _i}$ such that

$$ \begin{align*} [X_i,Y_i] = H_\epsilon-H_{i},\quad\text{for }1\leq i\leq n, \end{align*} $$

$$ \begin{align*}[X_{n+i},Y_{n+i}] = -H_\epsilon-H_{n-i+1},\quad\text{for }1\leq i\leq n. \end{align*} $$

We observe the following crucial fact:

$$ \begin{align*} &\mathfrak n^+_{\bar 0} Y_i \subset \bigoplus_{i< j}{\mathbb C} Y_j,~[X_i,Y_j] \subset \mathfrak n^+_{\bar 0}, \end{align*} $$

for any $1\leq i<j\leq 2n.$ As a consequence, we have

(6-3) $$ \begin{align} X_iY_{j}\cdots Y_{2n}v =0, \end{align} $$

for any $1\leq i<j\leq 2n$ .

Let $\rho : =-n \epsilon + \sum _{1\leq i\leq n} (n-i+1) \epsilon _i$ be the Weyl vector, that is, $\rho $ is the half of the sum of roots in $\Phi _{\bar 0}^+$ minus the half of the sum of roots in $\Phi ^+_{\bar 1}$ . Because $\lambda $ is atypical,

$$ \begin{align*} \prod_{\alpha\in\Phi^+_{\bar{1}}}(\lambda+\rho,\alpha) =0, \end{align*} $$

by [Gor1, Section 4.2]. Therefore, there exists $\alpha = \epsilon +c\delta _i$ such that $(\lambda +\rho ,\alpha ) =0$ and $c=\pm 1.$ We let v be a highest weight vector of $\Delta ^{\mathfrak {p}}(\lambda )$ . Also, we consider the element $\lambda = \lambda _\epsilon \epsilon +\sum _{1\leq i\leq n} \lambda _i \delta _i$ .

Suppose that $c=1$ . By (6-3) and direct computation, we get $X_iY_{i+1}\cdots Y_{2n}v =0$ and

$$ \begin{align*} &X_iY_iY_{i+1}\cdots Y_{2n}v \\[3pt] &\quad= (H_\epsilon -H_i)Y_{i+1}\cdots Y_{2n}v \\[3pt] &\quad= (\lambda_\epsilon -\lambda_i-2n+i -1) Y_{i+1}\cdots Y_{2n}v \\[3pt] &\quad=(\lambda+\rho, \alpha)Y_{i+1}\cdots Y_{2n}v =0. \end{align*} $$

Because the weight subspace of $\Delta ^{\mathfrak {p}}(\lambda )$ of weight $\lambda -\sum _{p\leq q\leq 2n}\alpha _q$ is one-dimensional and spanned by $Y_{p}Y_{p+1}\cdots Y_{2n}v$ , for any $1\leq p\leq 2n$ , we may conclude that $Y_{i+1}\cdots Y_{2n}v \notin X_i\Delta ^{\mathfrak {p}}(\lambda )$ , for otherwise $Y_{i+1}\cdots Y_{2n}v=0$ , which is a contradiction. Consequently, $X_{\Delta ^{\mathfrak {p}}(\lambda )}\neq 0$ . Hence, in this case, the fact that the associated variety of $\Delta ^{\mathfrak {p}}(\lambda )$ is nonzero follows from Corollary 26.

Suppose that $c=-1$ . Set $\overline i:=n-i+1$ for each $1\leq i\leq n$ . Similarly, using (6-3) and direct computation, we obtain $X_{n+\overline i}Y_{n+\overline i +1 +1}Y_{2n}v=0$ and

$$ \begin{align*} &X_{n+\overline i}Y_{n+\overline i}Y_{n+\overline i+1}\cdots Y_{2n}v \\[3pt] &\quad= -(H_\epsilon +H_i)Y_{n+\overline i+1}\cdots Y_{2n}v \\[3pt] &\quad=(-1) (\lambda_\epsilon +\lambda_i -i+1) Y_{n+\overline i+1}\cdots Y_{2n}v \\[3pt] &\quad=-(\lambda+\rho, \alpha)Y_{n+\overline i+1}\cdots Y_{2n}v =0. \end{align*} $$

As above, this implies that $Y_{n+\overline i+1}\cdots Y_{2n}v \notin X_{n+\overline i}\Delta \mathfrak {p}(\lambda )$ and yields $X_{\Delta ^{\mathfrak {p}}(\lambda )}\neq 0$ . Again, the fact that the associated variety of $\Delta ^{\mathfrak {p}}(\lambda )$ is nonzero now follows from Corollary 26. This completes the proof.

Corollary 35. Suppose that $\lambda \in \Sigma _{\mathfrak {p}}^+$ . Then the following are equivalent:

1. (1) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}\nabla ^{\mathfrak {p}}(\lambda ) <\infty $ ;

2. (2) ${\mathrm {pd}}_{\mathcal O}\nabla ^{\mathfrak {p}}(\lambda ) <\infty $ ;

3. (3) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}\nabla ^{\mathfrak {p}}(\lambda ) = {\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}_{\bar 0}}\nabla ^{\mathfrak {p}}(\lambda ) = {\mathrm {pd}}_{\mathcal O}\nabla ^{\mathfrak {p}}(\lambda ) -2\ell (w_0^{\mathfrak {p}})$ ;

4. (4) $\lambda $ is typical.

Proof. The implication $(1)\Rightarrow (3)$ follows from [Co, Corollary 3.2.5] and [Co, Lemma 6.3.2]. Therefore, the statements in $(1)$ and $(3)$ are equivalent. The numbers in part (3) are finite by [BGG2, Section 7].

We recall that $\mathfrak g= \mathfrak {osp}(2|2n)$ is one of the contragredient Lie superalgebras in [Mu, Theorem 5.1.5]. Therefore, $\mathcal O^{\mathfrak {p}}$ admits a simple-preserving duality (see, for example, [Mu, Section 13.7]) which sends dual Verma supermodules to Verma supermodules. By [Ma3, Theorem 3], it follows from Proposition 34 that the statements in $(1),(2)$ , and $(4)$ are equivalent.

6.2 Example II: projective dimension of modules for $\mathfrak {pe} (n)$

In this section, we assume that $\mathfrak g=\mathfrak {pe} (n)$ , for some positive integer n. It is known from [CCC, Section 5.5] that every parabolic category is equivalent to the parabolic category given by a parabolic subalgebra $\mathfrak p$ with $\mathfrak p_{\bar 1}=\mathfrak g_1$ . We recall that $\Sigma ^+_{\mathfrak {p}}$ denotes the set of highest weights of simple modules in $\mathcal O^{\mathfrak {p}}$ .

Proposition 36. Suppose that $M\in \mathcal O^{\mathfrak {p}}$ is a nonzero quotient of $\Delta ^{\mathfrak {p}}(\lambda )$ , for some $\lambda \in \Sigma ^+_{\mathfrak {p}}$ . Then the associated variety $X_M$ is nonzero. In particular,

$$ \begin{align*}{\mathrm{pd}}_{\mathcal O}M ={\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}}M =\infty.\end{align*} $$

Especially, the modules $L(\lambda ),~K(\lambda )$ and $\Delta ^{\mathfrak {p}}(\lambda )$ all have infinite projective dimension in both $\mathcal O$ and $\mathcal O^{\mathfrak {p}}$ .

Proof. We may observe that the element $2\epsilon _n$ is a root of $\mathfrak g_1$ but $-2\epsilon _n$ is not a weight of $\Lambda ^{\bullet } \mathfrak g_{-1}\otimes U(\mathfrak n^-_{\bar 0})$ . The assertion of the proposition now follows directly from Corollary 29.

We recall from [Se2] that a weight $\lambda =\sum _{1\leq i\leq n}\lambda _i\epsilon _i\in \mathfrak {h}^\ast $ is called atypical if

$$ \begin{align*} \prod_{1\leq i\neq j \leq n}(\lambda_i -\lambda_j +j-i-1) =0. \end{align*} $$

We describe the projective dimension of costandard objects in the following theorem.

Theorem 37. Let $\lambda \in \Sigma ^+_{\mathfrak {p}}$ . Then the following conditions are equivalent:

1. (1) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}} \nabla ^{\mathfrak {p}}(\lambda ) <\infty ;$

2. (2) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}} \nabla ^{\mathfrak {p}}(\lambda ) ={\mathrm {pd}}_{\mathcal O_{\bar 0}^{\mathfrak {p}}} \nabla ^{\mathfrak {p}}_{\bar 0}(\lambda );$

3. (3) $\lambda $ is typical.

Proof. From [Co, Corollary 3.2.5] and [Co, Lemma 6.3.2], it follows that

$$ \begin{align*}{\text{pd}}_{\mathcal O^{\mathfrak{p}}}{\nabla}^{\mathfrak{p}}_{\mathcal O^{\mathfrak{p}}}(\lambda) = {\text{pd}}_{\mathcal O^{\mathfrak{p}}_{\bar 0}}{\nabla}^{\mathfrak{p}}_{\bar 0}(\lambda),\end{align*} $$

if ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}\nabla ^{\mathfrak {p}}(\lambda )<\infty .$ Therefore, it remains to show that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}{\nabla }^{\mathfrak {p}}(\lambda )<\infty $ if and only if $\lambda $ is typical.

We first suppose that $\lambda $ is typical. Then, by [CP, Theorem 4.6],

$$ \begin{align*}(T^{\mathfrak{p}}(\lambda): \nabla^{\mathfrak{p}}(\mu)) = (T^{\mathfrak{p}}_{\bar 0}(\lambda): \nabla^{\mathfrak{p}}_{\bar 0}(\mu)),\end{align*} $$

for any $\mu \in \Sigma _{\mathfrak {p}}^+$ . We start by fixing $\lambda $ such that $T^{\mathfrak {p}}(\lambda ) = \nabla ^{\mathfrak {p}}(\lambda )$ . That is, let $J^{\mathfrak {p}}_\lambda $ be the intersection of the set of shortest representatives in $W^{\mathfrak {p}} \backslash W$ and the set of longest representatives in $ W/W_\lambda $ . We choose $\lambda $ that is a minimal element in $J^{\mathfrak {p}}_\lambda \cdot \lambda $ . We want to show that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}} \nabla (\mu ) <\infty $ , for any $\mu \in J^{\mathfrak {p}}_\lambda \cdot \lambda $ .

We first note that

$$ \begin{align*} &{\text{pd}}_{\mathcal O^{\mathfrak{p}}}\nabla^{\mathfrak{p}}(\lambda) ={\text{pd}}_{\mathcal O^{\mathfrak{p}}} T^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} T^{\mathfrak{p}}_{\bar 0}(\lambda) = {\text{pd}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} \nabla^{\mathfrak{p}}_{\bar 0}(\lambda), \end{align*} $$

by Theorem 20. Assume that $x\in J^{\mathfrak {p}}_\lambda $ is such that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}\nabla (y\cdot \lambda )<\infty ,$ for any $y\in J^{\mathfrak {p}}_\lambda $ with $x\cdot \lambda> y\cdot \lambda .$ By [CCC, Theorem 3.5], there is a submodule K of $T^{\mathfrak {p}}(x\cdot \lambda )$ such that we have the short exact sequence

$$ \begin{align*}0\rightarrow K\rightarrow T^{\mathfrak{p}}(x\cdot \lambda) \rightarrow \nabla^{\mathfrak{p}}(x\cdot \lambda) \rightarrow 0,\end{align*} $$

where $(K:\nabla ^{\mathfrak {p}}(z\cdot \lambda ))>0$ only if $z\in J^{\mathfrak {p}}_\lambda $ and $x\cdot \lambda>z\cdot \lambda $ . From our assumption, we have ${\text {pd}} K<\infty $ . Because ${\text {pd}}_{\mathcal O^{\mathfrak {p}}} T^{\mathfrak {p}}(x\cdot \lambda )<\infty $ , it follows that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}} \nabla ^{\mathfrak {p}}(x\cdot \lambda )<\infty $ . This completes the proof of the implication $(3)\Rightarrow (1).$

Before proving the direction $(1)\Rightarrow (3)$ , we make the following observations.

For any positive odd root $\alpha = \epsilon _i +\epsilon _j$ , we let $X_\alpha \in \mathfrak {g}_1$ be a root vector for $\alpha $ . If $\alpha =\epsilon _i +\epsilon _j$ with $i< j$ , we can pick root vectors $Y_{\alpha }\in \mathfrak {g}_{-1}$ for odd roots $-\alpha $ such that $[Y_{\alpha }, X_{\alpha }] =H_i-H_j$ .

Define a total ordering $\preceq $ on the set $\Phi _{\bar 1}^+:=\{\epsilon _i+\epsilon _j\mid 1\leq i<j\leq n\}$ by letting

$$ \begin{align*}\epsilon_i +\epsilon_{j+1}\prec \epsilon_i+\epsilon_{j},~2\epsilon_{i}\prec \epsilon_{i-1} +\epsilon_n,\end{align*} $$

for $1< i\leq n$ and $1\leq j<n$ . For any $1\leq i<j\leq n$ and any subset

$$ \begin{align*}I\subset \hat I:=\{\epsilon_i+\epsilon_t\mid i\leq t\leq j-1\},\end{align*} $$

we define

$$ \begin{align*} &S_{i,j,I}:=\{\epsilon_s+\epsilon_t\mid s<t,~1\leq s\leq i-1,~1\leq t\leq j\}\cup I. \end{align*} $$

Let $\lambda \in \mathfrak {h}^\ast $ and $v_\lambda \in \nabla ^{\mathfrak {p}}_{\bar 0}(\lambda )\subset {\nabla }^{\mathfrak {p}}(\lambda )$ be a highest weight vector. Set

$$ \begin{align*}X^{i,j,I}:=\prod_{\alpha \in S_{i,j,I}}X_\alpha\otimes v_\lambda \in {\nabla}^{\mathfrak{p}}(\lambda).\end{align*} $$

We observe:

1. (a) for any $\alpha \in \Phi _{\bar 1}^+,$ we have

   $$ \begin{align*}[X_\alpha,\mathfrak n_{\bar 0}^+] \subseteq \bigoplus_{\beta\in\Phi_{\bar 1}^+, ~\alpha \prec \beta}{\mathbb C} X_\beta;\end{align*} $$

2. (b) $[Y_{\epsilon _i+\epsilon _j}, X_{\epsilon _k+\epsilon _i}]\in \mathfrak n_{\bar 0}^+$ has weight $\epsilon _k-\epsilon _j$ , for $1\leq k\leq i$ ;

3. (c) $[Y_{\epsilon _i+\epsilon _j}, X_{\epsilon _i+\epsilon _k}]\in \mathfrak n_{\bar 0}^+$ has weight $\epsilon _k-\epsilon _j$ , for $i<k<j$ ;

4. (d) $[Y_{\epsilon _i+\epsilon _j}, X_{\epsilon _k+\epsilon _j}]\in \mathfrak n_{\bar 0}^+$ has weight $\epsilon _k-\epsilon _i$ , for $1\leq k<i$ .

As a consequence, from (a)–(d) above, it follows that

$$ \begin{align*} &Y_{\epsilon_i +\epsilon_j} X^{i,j,I}=0. \end{align*} $$

We can now proceed with the proof of $(1)\Rightarrow (3)$ . Suppose that $\lambda $ is integral and atypical with $\lambda _i -\lambda _j +j-i=\pm 1$ , for some $i<j$ , satisfying ${\text {pd}}{\nabla }^{\mathfrak {p}}(\lambda )<\infty $ . Then, by Corollary 26, the associated variety $X_{\nabla ^{\mathfrak {p}}(\lambda )}$ is zero.

We have to consider the following two cases.

Case 1. Suppose that $\lambda _i -\lambda _j +j-i+1=0$ .

Let $x:=X^{i,j,\hat I}$ and $y:=Y_{\epsilon _i+\epsilon _j}$ . Then $X_{\nabla ^{\mathfrak {p}}(\lambda )}=0$ implies that ${\nabla }^{\mathfrak {p}}(\lambda )_y=0$ . Therefore, $x\in y{\nabla }^{\mathfrak {p}}(\lambda )$ .

Assume that $m\in {\nabla }^{\mathfrak {p}}(\lambda )$ is a weight vector such that $yx=0$ and $x=ym$ . Then $m \in {\mathbb C} X_{\epsilon _i+\epsilon _j}X^{i,j,\hat I}.$ This implies that

$$ \begin{align*}x=ym\in {\mathbb C} Y_{\epsilon_i+\epsilon_j}X_{\epsilon_i+\epsilon_j}X^{i,j,\hat I} = {\mathbb C}(H_i -H_j)X^{i,j,\hat I} = \lambda_i -\lambda_j +j-i+1 =0, \end{align*} $$

a contradiction.

Case 2. Suppose that $\lambda _i -\lambda _j +j-i-1=0$ . Set $I:= \hat I \backslash \{2\epsilon _i\}$ .

Let $x:=X^{i,j, I}$ and $y:=Y_{\epsilon _i+\epsilon _j}$ . Then $X_{\nabla ^{\mathfrak {p}}(\lambda )}=0$ implies that ${\nabla }^{\mathfrak {p}}(\lambda )_y=0$ . Again, we have $x\in y{\nabla }^{\mathfrak {p}}(\lambda )$ .

Assume that $m\in {\nabla }^{\mathfrak {p}}(\lambda )$ is a weight vector such that $yx=0$ and $x=ym$ . Observe that

$$ \begin{align*}(H_i-H_j)m = (\lambda_i-\lambda_j +j-i-1)m=0,\end{align*} $$

and

$$ \begin{align*}m = X^{i,j,\hat I}\otimes m_j+ \sum_{i\leq t< j} X^{i,j,\hat I\backslash \{\epsilon_i+\epsilon_t\}\cup \{\epsilon_i+\epsilon_j\}}\otimes m_t,\end{align*} $$

for some $m_j, m_t \in {\nabla }_{\bar 0}(\lambda ).$ By (a)–(d), for $i\leq t <j$ , we have

$$ \begin{align*}yX^{i,j,\hat I}\otimes m_j =0,\end{align*} $$

$$ \begin{align*}y X^{i,j,\hat I\backslash \{\epsilon_i+\epsilon_t\}\cup \{\epsilon_i+\epsilon_j\}}\otimes m_t =(H_i-H_j)X^{i,j,\hat I\backslash \{\epsilon_i+\epsilon_t\}}\otimes m_t=0.\end{align*} $$

The obtained contradiction completes the proof.